rough burgers-like equations with multiplicative …space of stochastic processes to deal with the...

ROUGH BURGERS-LIKE EQUATIONS WITH MULTIPLICATIVE NOISE

MARTIN HAIRER AND HENDRIK WEBER

ABSTRACT. We construct solutions to Burgers type equations perturbed by a multiplicative space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are notregular enough to be amenable to classical methods. We use the theory of controlled rough paths togive a meaning to the spatial integrals involved in the definition of a weak solution. Subject to thechoice of the correct reference rough path, we prove unique solvability for the equation and we showthat our solutions are stable under smooth approximations of the driving noise.

1. INTRODUCTION

The main goal of this article is to first provide a good notion of a solution and then to establishtheir existence and uniqueness for systems of stochastic partial differential equations of the form

du =[∆u+ g(u)∂xu

]dt+ θ(u) dW (t), (1.1)

where u : [0, T ]× [0, 1]→ Rn. Here, the functions g : Rn → Rn×n as well as θ : Rn → Rn×n areassumed to be sufficiently smooth and the operator ∆ = ∂2

x denotes the Laplacian with periodicboundary conditions on [0, 1] acting on each coordinate of u. Finally, W (t) is a cylindrical Wienerprocess on L2([0, 1],Rn), i.e. we deal with space-time white noise.

Our motivation for studying (1.1) is twofold. On the one hand, similar equations, but with additivenoise, arise in the context of path sampling, when the underlying diffusion has additive noise [18, 15].While we expect the case of underlying diffusions with multiplicative noise to be more complex thanthe equations considered in this article, we believe that it already provides a good stepping stonefor the understanding of SPDE with rough solutions and state-dependent noise. Another long-termmotivation is the understanding of the solutions to the one-dimensional KPZ equation [22, 3] andtheir construction without relying on the Cole-Hopf transform. While that equation might at firstsight be quite remote from the problem at hand (it has additive noise and the nonlinearity has adifferent structure), it is possible to perform formal manipulations of its solutions that yield equationswith features that are very close to those of the equations considered in the present work.

The main obstacle we have to overcome lies in the spatial regularity of the solutions. Actually,solutions to the linear stochastic heat equation

dX = ∆X dt+ dW (t), (1.2)

are not differentiable as a function of (t, x) – X is almost surely α-Holder continuous for everyα < 1

2 as a function of the space variable x and α2 -Holder continuous as a function of the time

variable t. However, it is almost surely not 12 -Holder continuous as a function of x. Since we assume

u to have similar regularity properties, it is a priori not clear how to interpret the spatial derivative inthe nonlinear term g(u)∂xu.

So far, equation (1.1) has mostly been studied under the assumption that there exists a functionG such that g = DG (see e.g. [14, 3]). Clearly in the one-dimensional case n = 1 such a function

Date: December 6, 2010.

1

2 MARTIN HAIRER AND HENDRIK WEBER

always exists whereas in the higher-dimensional case this is not true. Then weak solutions can bedefined as processes satisfying

d〈ϕ, u〉 =[〈∆ϕ, u〉 − 〈∂xϕ,G(u)〉

]dt+ 〈ϕθ(u),dW (t)〉, (1.3)

for every smooth periodic test function ϕ. It is known [14] that under suitable regularity and growthassumptions on G and θ, such solutions can then indeed be constructed and are unique. When sucha primitive G does not exist, this method cannot be applied and a concept of solutions has not yetbeen provided.

Recently, in [15] a new approach was proposed to deal with this problem in the additive noisecase θ = 1. There the non-linearity in (1.3) is rewritten as

〈ϕ, g(u)∂xu〉 =

∫ 1

0ϕ(x) g

(u(x)

)dxu(x). (1.4)

The (spatial) regularity of u is not sufficient to make sense of this integral in a pathwise sense usingYoung’s integration theory. This means in particular that extra stochastic cancellation effects arenecessary to give a meaning to (1.4). In [15] these problems are treated using Lyons’ rough paththeory (see [25, 12, 10]).

In this approach the definition of integrals like (1.4) is separated into two steps: First a referencerough path has to be constructed i.e. a pair of stochastic process (X,X) satisfying a certain algebraiccondition. The process X should be thought of as the iterated integral

X(x, y) =

∫ y

x

(X(z)−X(x)

)⊗ dX(z). (1.5)

It is usually constructed using a stochastic integral like the Ito or Stratonovich integral. If on smallscales u behaves like X (see Section 2 for a precise definition of controlled rough path) integralslike 1.4 can be defined as continuous functions of their data.

The advantage of this approach is that the stochastic cancellation effects are captured in thereference rough path and can be dealt with independently of the rest of the construction. As areference rough path the solution X of the linear heat equation (1.2) is chosen. As this process isGaussian known existence and continuity results [9, 10] rough paths can be applied.

This article provides an extension of [15] to the multiplicative noise case θ 6= 1. Roughlyspeaking, the construction of the additive noise case is extended by another fixed point argument.One tricky part is that it is no longer clear how to interpret solution to the “linear” heat equation

dΨθ = ∆u+ θdW (1.6)

as rough path valued processes since, if θ is a stochastic process, Ψθ is not Gaussian in general andso the results of [9, 10] do not apply directly. We resolve this issue by showing that as soon as θ hassufficient space-time regularity, the process Ψθ can be interpreted for every fixed t > 0 as a roughpath (in space) controlled by the rough path X constructed from solutions to (1.2). We are able touse this knowledge in Definition 3.1 below to formulate what we actually mean by a solution to(1.1). Such solutions are then obtained by combining an (inner) fixed point argument in a space ofdeterministic functions to deal with the non-linearity and another (outer) fixed point argument in aspace of stochastic processes to deal with the multiplicative noise.

The solutions we construct in this way depend on the choice of the reference rough path (X,X).Since there is a priori some freedom in the definition of the iterated integral (1.5), similar to thechoice between Ito or Stratonovich integral, we can get several possible solutions. This will bediscussed at the end of Section 3. Actually, these different solutions are not only an artefact of theclassical ill-posedness of our equation, since they also appear in the gradient case g = DG. Thismay sound surprising, but in a series of recent works [16, 19, 17], approximations to stochastic

ROUGH BURGERS 3

equations with similar regularity properties as (1.1) were studied. It was shown there that severalseemingly natural approximation schemes involving different approximations of the non-linearterm may produce non-trivial correction terms in the limit, that correspond exactly to this kind ofIto-Stratonovich correction. In the gradient case, this ambiguity can be removed by imposing theadditional assumption that the reference rough path is geometric (see Section 3 for a discussion).

In the non-gradient case however, this is not sufficient to characterise the solution uniquely. Evenamong the geometric reference rough paths, different pure area type terms may appear. We arguethat the “canonical” construction of X given in [9, 10] is still natural for the problem at hand since,if we replace the noise by a mollified version and then send the scale ε of the mollifier to 0, we showthat the corresponding sequence of solutions converges to the solution constructed in this article.Note that the fact that this sequence of solutions even remains bounded as ε→ 0 does not followfrom standard bounds. In particular, we would not know a priori how to use these approximationsto construct solutions to (1.1). These convergence results do not contradict the appearance of theadditional correction terms discussed in [16, 19, 17], since the approximations considered there alsoinvolve an approximation of the nonlinear term g(u)dxu.

Of course our work is by no means the only work that establishes an application of rough pathmethods to the theory of stochastic PDE. Recently Gubinelli and Tindel [13] used a rough pathapproach to construct mild solutions to equations of the type

du = Audt+ F (u) dW (1.7)

for a rough path W taking values in some space of possibly quite irregular functions. However,in the case of A being the Laplacian on [0, 1] an F being a composition operator, they requiredthe covariance of the noise to decay at least like ∆−1/6, thus ruling out space-time white noise.Teichmann [29] suggests another approach using the method of the moving frame, but these resultsdeal with a different class of equations and are designed to deal with noise terms that are rough intime, but rather smooth in space. Yet another approach to treat equations of the type

du(t, x) = F (t, x,Du,D2u) dt−Du(t, x)V (x) dW (t) (1.8)

using the stochastic method of characteristics is presented by Caruana, Friz, and Oberhauser in [4],extending ideas from [23]. In all of the above works, rough path theory is used to define the temporalintegrals. To our knowledge [15] was the first article to make use of rough integrals to deal withspatial regularity issues.

There is also a wealth of literature on the problem of how to treat stochastic PDEs with solutionsthat are very rough in space. A large part of it is inspired by the ideas of renormalisation theorycoming from quantum field theory. For example, it was possible in [20, 1, 7, 6] to rigorouslyconstruct solutions to the stochastic Allen-Cahn and Navier-Stokes equations in two spatial di-mensions, driven by space-time white noise. It is not clear whether these techniques apply to ourproblem. Furthermore, our results allow us to obtain very fine control on the solutions and on theconvergence of approximations, on the contrary of the above mentioned works, where solutions areonly constructed for a set of initial conditions that is of full measure with respect to the invariantmeasure. A related but somewhat different approach is to use Wick calculus in Wiener space as in[2, 28, 5], but this leads to different equations, the interpretation of which is not clear.

The remainder of the article is structured as follows: In Section 2 we give a brief account ofthose notions of rough path theory that we will need. We follow Gubinelli’s approach [12] to definerough integrals and recall the existence and continuity results for Gaussian rough path from [9]. InSection 3 we give a rigorous definition of our notion of a solution to (1.1) and we state the mainresults of this work. This section also contains a discussion of the dependence of solutions on thechoice of reference rough path. In Section 4, we then discuss the stochastic convolution Ψθ. In


particular, we show that as soon as θ possesses the right space-time regularity it is controlled by thelinear Gaussian stochastic convolution. The proof of the main results is then given in Section 5. Thebeginning of this section also contains a sketch of the main “two-level” fixed point argument uponwhich our proofs rely.

1.1. Notation. We will deal with functions u = u(t, x;ω) depending on a time and a space variableas well as on randomness. Norms that only depend on the behaviour of u as a function of space forfixed t and ω will be denoted with | · |, norms that depend on the behaviour as a function of (t, x) forfixed ω with ‖ · ‖ and norms that depend on all parameters with ||| · |||.

We will denote by C a generic constant that may change its value at every occurrance.

Acknowledgements. We are grateful to Peter Friz, Massimilliano Gubinelli, Terry Lyons, Jan Maas, AndrewStuart, and Jochen Voß for several fruitful discussions about this work. Financial support was kindly providedby the EPSRC through grant EP/D071593/1, as well as by the Royal Society through a Wolfson ResearchMerit Award and by the Leverhulme Trust through a Philip Leverhulme Prize.

2. ROUGH PATHS

In this section we recall the elements of rough path theory which we will need in the sequel andrefer the reader to e.g. [10, 12, 24, 26] for a more complete account. We introduce Gubinelli’snotion of a controlled rough path [12] and give the main existence and continuity statements forGaussian rough path from [9].

For a normed vector space V we denote by C(V ) the space of continuous functions from [0, 1] toV and by ΩC(V ) the space of continuous functions from [0, 1]2 to V which vanish on the diagonal(i.e. for R ∈ ΩC(V ) we have R(x, x) = 0 for all x ∈ [0, 1]). We will often omit the reference tothe space V and simply write C and ΩC.

It will be useful to introduce for X ∈ C and R ∈ ΩC the operators

δX(x, y) = X(y)−X(x) (2.1)

andNR(x, y, z) = R(x, z)−R(x, y)−R(y, z). (2.2)

The operator δ maps C into ΩC and N δ = 0. The quantity NR can be interpreted as an indicatorto how far R is from the image of δ.

In the sequel α will always be a parameter in(

13 ,

12

). For X ∈ C and R ∈ ΩC we define

Holder-type semi-norms:

|X|α = supx 6=y

|δX(x, y)||x− y|α

and |R|α = supx 6=y

|R(x, y)||x− y|α

. (2.3)

We denote by Cα resp. ΩCα the set of functions for which these semi-norms are finite. The spaceCα endowed with | · |Cα = | · |0 + | · |α is a Banach-space. Here | · |0 denotes the supremum norm.The space ΩCα(V ) is a Banach space endowed with | · |α alone.

Remark 2.1. One might feel slightly uneasy working in these Holder spaces as they are notseparable. We will neglect this issue noting that all the processes we will consider will actually takevalues in the slightly smaller space of Cα-functions that can be approximated by smooth functionswhich is separable.

Definition 2.2. An α-rough path on [0, 1] consists of a pairX ∈ Cα(Rn) and X ∈ ΩC2α(Rn⊗Rn

)satisfying the relation

NXij(x, y, z) = δXi(x, y)δXj(y, z) (2.4)

ROUGH BURGERS 5

for all 0 ≤ x ≤ y ≤ z ≤ 1 and all indices i, j ∈ 1, . . . , n. We will denote the set of α-roughpaths by Dα(Rn) or simply by Dα.

As explained above X(x, y)ij - called the iterated integral should be interpreted as the value of

X(x, y)ij =

∫ y

x

(Xi(z)−Xi(x)

)dXj(z). (2.5)

If X is smooth enough for (2.5) to make sense it is straightforward to check that X(x, y)ij definedby (2.5) satisfies (2.4). On the other hand we do not require consistency in the sense that X(x, y)ij

needs not be defined by (2.5), even if it would make sense. Due to the non-linear constraint (2.4) theset of α-rough paths is not a vector space. But it is a subset of the Banach space Cα ⊕ ΩCα and wewill use the notation

|(X,X)|Dα = |X|Cα + |X|2α. (2.6)Let us recall the basic existence and continuity properties for Gaussian rough paths. Following

[9] we define for ρ ≥ 1 the 2-dimensional ρ-variation of a function K : [0, 1]2 → R in the cube[x1, y1]× [x2, y2] ⊆ [0, 1]2 as

|K|ρ−var([x1, y1]× [x2, y2]

)=(

sup∑i,j

∣∣K(zi+11 , zj+1

2 ) +K(zi1, zj2)−K(zi1, z

j+12 )−K(zi+1

1 , zj2)∣∣ρ) 1

ρ, (2.7)

where the supremum is taken over all finite partitions x1 ≤ z11 ≤ . . . ≤ zm1

1 = y1 and x2 ≤ z12 ≤

. . . ≤ zm22 = y2

This notion of finite two-dimensional 1-variation does not coincide with the classical conceptof being of bounded variation. Actually a function is BV if its gradient is a vector valued measure,whereas K is of finite 2-dimensional 1-variation if ∂x∂yK is a measure.

The following lemma is a slightly modified version of the existence and continuity results from[9]:

Lemma 2.3 ([9, Thm 35] and [10, Cor. 15.31]). Assume that X = (X1(x), . . . , Xn(x)), x ∈ [0, 1]is a centred Gaussian process. We assume that the components are independent of each other anddenote by Ki(x, y) the covariance function of the i-th component.

Assume that there the KXi satisfy the following bound for every 0 ≤ x ≤ y ≤ 1

|KXi |ρ−var

([x, y]2

)≤ C|y − x|. (2.8)

Then for every α < 12ρ the process X can canonically be lifted to an α rough path (X,X) and for

every p ≥ 1

E[|X|pα

]<∞ and E

[|X|p2α

]<∞. (2.9)

Furthermore, let Y = (Y1(x), . . . , Yn(x)) be another process such that (X,Y ) is jointly Gaussianand (Xi, Yi) and (Xj , Yj) are mutually independent for i 6= j. Assume that the covariance functionsKXi and KY

i satisfy (2.8) and that

|KX−Yi |ρ−var

([x, y]2

)≤ Cε2|y − x|

1ρ , (2.10)

then for every pE[|X−Y|p2α

]1/p ≤ Cε. (2.11)

In this context canonically lifted means that X is constructed by considering approximationsto X , defining approximate iterated integrals using (2.5) and then passing to the limit. Severalapproximations including piecewise linear, mollification and a spectral decomposition yield the


same result. In the case where X is an n-dimensional Brownian motion, X coincides with theStratonovich integral.

Note that the condition (2.8) implies that the classical Kolmogorov criterion

E[|X(x)−X(y)|2

]≤ C|x− y|

1ρ , (2.12)

for α-Holder continuous sample paths is satisfied.

Remark 2.4. The results in [9, 10] imply more than we state in Lemma 2.3. In particular, thereit is proven that (X,X) is a geometric rough path. We will not discuss the concept of geometricrough path in detail (see e.g. [10] for the geometric aspects of rough path theory), but remark that itimplies that for every x, y the symmetric part of X(x, y) is given as

Sym(X(x, y)

)=

1

2

(X(x, y) + X(x, y)T

)=

1

2δX(x, y)⊗ δX(x, y). (2.13)

Our goal is to define integrals against rough paths. Here Gubinelli’s approach seems to be bestsuited to our needs. Thus following [12] we define:

Definition 2.5. Let (X,X) be in Dα(Rn). A pair (Y, Y ′) with Y ∈ Cα(Rm) and Y ′ ∈ Cα(Rm ⊗Rn) is said to be controlled by (X,X) if for all 0 ≤ x ≤ y ≤ 1

δY (x, y) = Y ′(x)δX(x, y) +RY (x, y), (2.14)

with a remainder RY ∈ ΩC2α(Rm). Denote the space of Rm-valued controlled rough paths (X,X)by CαX(Rn) (or simply CαX ) .

Note that the constraint (2.14) is linear and in particular CαX is a vector space although Dα is not.We will use the notation

|(Y, Y ′)|CαX = |Y |Cα + |Y ′|Cα + |RY |2α (2.15)

for (Y, Y ′) ∈ CαX .In general the decomposition (2.14) needs not be unique but it is unique as soon as for every

x ∈ [0, 1] there exists a sequence xn → x such that

|X(x)−X(xn)||x− xn|2α

→∞, (2.16)

i.e. if X is rough enough. In most of the situations we will encounter, this will be the case almostsurely and there is a natural choice of Y ′. We will therefore often drop the reference to the derivativeY ′ and simply refer to Y as a controlled rough path.

Given two α-Holder paths Y, Z ∈ Cα(Rn) it is generally not possible to prove convergence ofthe Riemann sums ∑

i

Y (xi)⊗(Z(xi+1)− Z(xi)

)(2.17)

if the partition 0 ≤ x1 ≤ . . . ≤ xm = 1 becomes finer. If we know in addition that Y, Z arecontrolled by a rough path (X,X) it is natural to construct the integral

∫Y dZ by a second order

approximation ∑i

Y (xi)⊗(Z(xi+1)− Z(xi)

)+ Y ′(xi)X(xi, xi+1)Z ′(xi)

T . (2.18)

As we always assume α > 13 , it turns out that the approximations (2.18) converge:

ROUGH BURGERS 7

Lemma 2.6 ([12, Thm 1 and Cor. 2]). Suppose (X,X) ∈ Dα(Rn) and Y,Z ∈ CαX(Rm) for someα > 1

3 . Then the Riemann-sums defined in (2.18) converge as the mesh of the partition goes to zero.We call the limit rough integral and denote it by

∫Y (x)⊗ dZ(x).

The mapping (Y, Z) 7→∫Y ⊗ dZ is bilinear and we have the following bound:∫ y

xY (z)⊗ dZ(z) = Y (x)⊗ δZ(x, y) + Y ′(x)X(x, y)Z ′(x)T +Q(x, y), (2.19)

where the remainder satisfies∣∣Q∣∣3α≤C

[|RY |2α|Z|α + |Y ′|0|X|α|RZ |2α

+ |X|2α(|Y ′|α|Z ′|0 + |Y ′|0|Z ′|α

)+ |X|2α|Y ′|0|Z|α

]. (2.20)

Remark 2.7. In the simplest possible (one-dimensional) case where Z(x) = x, Y (x) is a C2

function and xi = iN the second order approximation (2.18) corresponds to∑

i

Y (xi)(xi+1 − xi

)+

1

2Y ′(xi)

(xi+1 − xi

)2, (2.21)

and the convergence of this approximation towards∫Y dx is of order N−2 instead of N−1 for the

simple approximation∑

i Y (xi)(xi+1 − xi

).

Remark 2.8. In most situations it will be sufficient to simplify the bounds (2.19) and (2.20) to∣∣∣∣ ∫ Y (z)⊗ dZ(z)

∣∣∣∣α

≤ C(

1 + |(X,X)|2Dα)|Y |CαX |Z|CαX , (2.22)

Note however that in the original bounds (2.19) and (2.20), no term including either the product|RY |2α|RZ |2α or the product |Y ′|α|Z ′|α appears. We will need this fact when deriving a prioribounds in Section 5.

The rough integral also possesses continuity properties with respect to different rough paths.More precisely we have:

Lemma 2.9 ([12] page 104). Suppose (X,X), (X, X) ∈ Dα and Y,Z ∈ CαX as well as Y , Z ∈ CαX

for some α > 13 . Then we get the following bound∣∣∣ ∫ Y (z)⊗ dZ(z)−

∫Y (z)⊗ dZ(z)

∣∣∣α

≤C(|Y |CαX + |Y |Cα

X

)(|Z|CαX + |Z|Cα

X

)(|X − X|Cα + |X− X|2α

)+ C

(|Z|CαX + |Z|Cα

X

)(1 +

∣∣(X,X)|Dα +∣∣(X, X)|Dα)

·(|Y − Y |Cα + |Y ′ − Y ′|Cα + |RY −RY |2α

)+ C

(1 +

∣∣(X,X)|Dα +∣∣(X, X)|Dα)(|Y |CαX + |Y |Cα

X

)·(|Z − Z|Cα + |Z ′ − Z ′|Cα + |RZ −RZ |2α

). (2.23)

This bound behaves as if the rough integral were trilinear in Y,Z, and X . Unfortunately, thisstatement makes no sense, as Dα is not a vector space, and Y and Y (resp. Z and Z) take values indifferent spaces CαX and Cα

X.

We will need the following Fubini type theorem for rough integrals:


Lemma 2.10. Let (X,X) be an α-rough path and Y,Z ∈ CαX . Furthermore, assume that f =f(λ, y), (λ, y) ∈ [0, 1]2 is a function which is continuous in λ and uniformly C1 in y. Denote byWλ(y) = f(λ, y)Y (y) and by W (y) =

( ∫ 10 f(λ, y) dλ

)Y (y). Then W as well as the Wλ are

rough path controlled by X and the following Fubini-type property holds true∫ 1

0

(∫ 1

0f(λ, x) dλ

)Y (x) dZ(x) =

∫ 1

0

(∫ 1

0f(λ, x) Y (x) dZ(x)

)dλ. (2.24)

Proof. From the assumptions on f it follows directly that W is an X-controlled rough path whosedecomposition is given as

δW (x, y) =

(∫ 1

0f(λ, x) dλ

)Y ′(x)δX(x, y) +

(∫ 1

0f(λ, x) dλ

)RY (x, y)

+

(∫ 1

0f(λ, x)− f(λ, y) dλ

)Y (y), (2.25)

and similarly for the Wλ. Then we get from the definition of the rough integral (2.18), (2.19) and(2.20) that∫ 1

0W (x) dZ(x)

= lim∑i

W (xi)δZ(xi, xi+1) +W ′(xi)X(xi, xi+1)Z ′(xi)T

= lim

∫ 1

0

(∑i

Wλ(xi)δZ(xi, xi+1) +W ′λ(xi)X(xi, xi+1)Z ′(xi)T)

dλ. (2.26)

Where the limit is taken as the mesh of the partition 0 ≤ x1 ≤ · · · ≤ xN ≤ 1 goes to zero. Asall the error estimates on the convergence of the limit are uniform in λ the integral and the limitscommute.

To end this section we quote a version of a classical regularity statement due to Garsia, Rodemichand Rumsey [11] which has proven to be very useful to give quantitative statements about theregularity of a random paths.

Lemma 2.11 (Lemma 4 in [12]). Suppose R ∈ ΩC. Denote by

U =

∫[0,1]2

Θ

(R(x, y)

p(|x− y|/4

))dx dy, (2.27)

where p : R+ → R+ is an increasing function with p(0) = 0 and Θ is increasing, convex, andΘ(0) = 0. Assume that there is a constant C such that

supx≤u≤v≤r≤y

∣∣∣NR(u, v, r)∣∣∣ ≤ Θ−1

(C

|y − x|2

)p(|y − x|/4), (2.28)

for every 0 ≤ x ≤ y ≤ 1. Then∣∣R(x, y)∣∣ ≤ 16

∫ |y−x|0

[Θ−1

(U

r2

)+ Θ−1

(C

r2

)]dp(r). (2.29)

for any x, y ∈ [0, 1].

This version of the Garsia-Rodemich-Rumsey Lemma is slightly more general than the usual one(see e.g. [10, Thm A.1]). The classical version treats the case of a functions f on [0, 1] taking values

ROUGH BURGERS 9

in a metric space. Then the same conclusion holds if one replaces R(x, y) with d(A(x), A(y)

)and

C = 0. In the case that we will mostly use Θ(u) = up and p(x) = xα+2/p. Then Lemma 2.11 states

|f |α ≤ C(∫

[0,1]2

d(f(x), f(y)

)p|x− y|αp+2

dx dy

)1/p

, (2.30)

which is essentially a version of Sobolev embedding theorem.

3. MAIN RESULTS

Now we are ready to discuss solutions of the equation (1.1). Anticipating Proposition 4.7 we willuse that the solution X to linear stochastic heat equation

dX = ∆Xdt+ dWt

can be viewed as a rough path valued process. To be more precise for every t there exists a canonicalchoice of X(t, ·, ·) ∈ ΩC2α such that the pair (X(t),X(t)) is a rough path in the space variable x.Furthermore the process t 7→ (X(t),X(t)) is almost surely continuous in rough path topology. Wewill use this observation to make sense of the non-linearity by assuming that for every t the solutionu(t, ·) is a rough path controlled by X(t, ·).

Definition 3.1. A weak solution to (1.1) is an adapted process u = u(t, x) that takes values inCα/2,α

([0, T ] × [0, 1]

)∩ L2([0, T ]; CαX) such that for every smooth periodic test function ϕ the

following equation holds

〈ϕ, u(t)〉 = 〈ϕ, u0〉+

∫ t

0〈∆ϕ, u(s)〉ds+

∫ t

0

(∫ 1

0ϕ(x)g

(u(s, x)

)dxu(s, x)

)ds

+

∫ t

0〈ϕ(x) θ

(u(s, x)

),dW (s)〉. (3.1)

Here by u ∈ L2([0, T ]; CαX) we mean that almost surely for every t the function u(t, ·) iscontrolled by the random rough path

(X(t, ·),X(t, ·)

)and t 7→ |u|Cα

X(t)is almost surely in L2. By

Cα/2,α([0, T ]× [0, 1]

)we denote the space of functions u with finite parabolic Holder norm

‖u‖Cα/2,α = sups 6=t,x6=y

|u(s, x)− u(t, y)||x− y|α − |s− t|α/2

+ ‖u‖0. (3.2)

The spatial integral in the third term on the right hand side of (3.1) is a rough integral. To bemore precise, as for every s the function u(s, ·) is controlled by X(s, ·) and g and ϕ are smoothfunctions, ϕg(u(s, ·)) is also controlled by X(s, ·). Thus the spatial integral can be defined as in(2.19),(2.20). This integral is the the limit of the second order Riemann sums (2.18) and thus inparticular measurable in s. For every s this integral is bounded by C|g|C2 |ϕ|C1 |u(s)|2Cα

X(s)(1 +

|(X(s),X(s))|Dα). So the outer integral can be defined as a Lebesgue integral.A priori this definition depends on the choice of initial condition for the Gaussian reference rough

path X . In our construction we will use zero initial data but this is not important. In the constructionof global solutions we will in fact concatenate the constructions obtained this way and thus restart Xat deterministic time steps. This does not alter the definition of solution: Indeed, if u(t) is controlledby X(t) with rough path decomposition given as

δu(t, x, y) = u′(t, x)δX(t, x, y) +Ru(t, x, y)


and the modified rough path X is given as the solution to the stochastic heat equation startet at zeroat time t0 then for t ≥ t0 one has

X(t, x, y) = X(t, x, y)− S(t− t0)X(t0).

Therefore, u has a rough path decomposition with respect to X which is given as

δu(t, x, y) = u′(t, x)δX(t, x, y) +Ru(t, x, y)− u′(t, x)δS(t− t0)X(t0)(x, y).

So u is also controlled by X . Due to standard regularisation properties of the heat semigroup

|u′(t)δS(t− t0)X(t0)|2α ≤ C|u′|0(t− t0)−α2 |X(t0)|α.

This is square integrable, and so u satisfies the regularity assumption with respect to this referencerough path as well. Another choice of reference rough path would be to take the stationary solutionto the stochastic heat equation

dX =(∆X − X

)dt+ dWt.

This would also yield the same solutions.Our notion of a weak solution has an obvious mild solution counterpart:

Definition 3.2. A mild solution to (1.1) is a process u such as in Definition 3.1 with equation (3.1)replaced by

u(x, t) =S(t)u0(x) +

∫ t

0

(∫ 1

0pt−s(x− y) g(u(s, x)) dyu(s, y)

)ds

+

∫ t

0S(t− s) θ(u(s)) dW (s)(x). (3.3)

Here S(t) = e∆t denotes the heat semigroup with periodic boundary conditions, given byconvolution with the heat kernel pt, acting independently on every coordinate. As above the spatialintegral involving the nonlinearity g is a rough integral.

Remark 3.3. In Section 5 we will see that the term involving the nonlinearity on the right hand sideof (3.3) is always C1 in space. Thus, using Proposition 4.8 one can see that the controlled roughpath decompositon of u(t, ·) is given as

δu(t, x, y) = θ(u(t, x)

)δX(t, x, y) +R(t, x, y). (3.4)

As expected the notions of weak and mild solution coincide:

Proposition 3.4. Every weak solution to (1.1) is a mild solution and vice versa.

Proof. The proof follows the lines of the standard proof (see e.g. [8] Theorem 5.4). We only need tocheck that the argument is still valid when we use rough integration.

Assume first that u is a mild solution to (1.1) and let us show that then (3.1) holds. To this end letϕ be a periodic C2 function. Testing u against ∆ϕ and integrating in time we get∫ t

0〈∆ϕ, u(s)〉 ds =

∫ t

0〈∆ϕ, S(s)u0〉 ds+

∫ t

0〈∆ϕ,Ψθ(u)(s)〉

+

∫ t

0

∫ s

0

(∫ 1

0∆ϕ(x)

∫ 1

0ps−τ (x− y) g(u(τ, x)) dyu(τ, y) dx

)dτ ds. (3.5)

ROUGH BURGERS 11

Let us treat the term involving the nonlinearity g on the right hand side of (3.5). Using Fubinitheorem for the temporal integrals and then the Fubini theorem for rough integrals (2.24) we canrewrite this term as∫ t

0

(∫ 1

0

[ ∫ t

τ

∫ 1

0∆ϕ(x) ps−τ (x− y) dx ds

]g(u(τ, x)) dyu(τ, y)

)dτ

=

∫ t

0

(∫ 1

0

[ ∫ 1

0ϕ(x) ps(x− y) dx− ϕ(y)

]g(u(τ, x)) dyu(τ, y)

)dτ. (3.6)

where we have used the definition of the heat semigroup in the first line. Similar calculations for theremaining terms in (3.5) (see [8] Theorem 5.4 for details) show that∫ t

0〈∆ϕ, S(s)u0〉ds = 〈ϕ, S(t)u0〉 − 〈ϕ, u0〉, (3.7)

and ∫ t

0〈∆ϕ,Ψθ(u)(s)〉ds = 〈ϕ,Ψθ(u)(s)〉〉 −

∫ t

0〈ϕ, θ(u)(s)dW (s)〉. (3.8)

Thus summarising (3.5) - (3.8) we get (3.1).In order to see the converse implication, first note that in the same way as in ([8] Lemma 5.5)

one can show that if u is a weak solution the following identity holds if u is tested against a smooth,periodic and time dependent test function ϕ(t, x)

〈ϕ(t), u(t)〉 = 〈ϕ(0), u0〉+

∫ t

0

⟨(∆ϕ(s) + ∂sϕ(s)

), u(s)

⟩ds+

∫ t

0〈ϕ(s) θ

(u(s)

), dW (s)〉

+

∫ t

0

(∫ 1

0ϕ(s, x)g

(u(s, x)

)dxu(s, x)

)ds (3.9)

Taking ϕ(t, x) = S(t−s)ξ(x) for a smooth function ξ and using the definition of the heat semigroupone obtains

〈u(t), ξ〉 = 〈S(t)ξ, u0〉+

∫ t

0〈S(t− s)ξ θ

(u(s)

),dW (s)〉

+

∫ t

0

(∫ 1

0

∫ 1

0pt−s(x− y) ξ(y) g

(u(s, x)

)dxu(s, x) dy

)ds. (3.10)

Thus using again the Fubini theorem for rough integrals as well as the symmetry of the heatsemigroup one obtains (3.3).

Weak solutions do indeed exist:

Theorem 3.5. Fix 13 < α < β < 1

2 and assume that u0 ∈ Cβ . Furthermore, assume that g is abounded C3 function with bounded derivatives up to order 3 and that θ is a bounded C2 functionwith bounded derivatives up to order 2. Then for every time interval [0, T ] there exists a uniqueweak/mild solution to (1.1).

The construction will be performed uin the next section using a Picard iteration. The statementremains valid, if one includes extra terms that are more regular, as for example a reaction term f(u)for a bounded C1 function f with bounded derivatives.

It is important to remark that although the rough path (X,X) does not explicitly appear in (3.1) itplays a crucial role in determining the solution uniquely. Let us illustrate this in the one-dimensionalcase.

We could define another rough path valued process (X, X) by setting

X(t, x, y) = X(t, x, y) + (x− y). (3.11)


This corresponds to introducing an Ito – Stratonovich correction term in the iterated integral. Then itis straightforward to check that (X, X) also satisfies (2.4) and (3.1) makes perfect sense if we viewthe space integral in the nonlinearity as a rough integral controlled by (X, X). Using the definitionof the rough integral in Lemma 2.6 and recalling (3.4) we get

∫ 1

0ϕ(x) g

(u(x)

)dxu(x)

= lim∑i

ϕ(xi) g(u(xi)

)(u(xi+1)− u(xi)

)+ ϕ(xi)g

′(u(xi))θ(u(xi)

)2X(xi, xi+1)

= lim∑i

ϕ(xi)g(u(xi)

)(u(xi+1)− u(xi)

)+ ϕ(xi)g

′(u(xi))θ(u(xi)

)2(X(xi, xi+1) + (xi+1 − xi)

)=

∫ 1

0ϕ(x)g

(u(x)

)dxu(x) +

∫ 1

0ϕ(x)g′(u(x)) θ

(u(x)

)2dx, (3.12)

where by∫

we denote the rough integral with respect to (X, X). The limit is taken as the mesh ofthe partition 0 ≤ x1 ≤ · · · ≤ xN ≤ 1 goes to zero.

In particular, the reaction term g′(u)θ(u)2 appears. In the additive noise case θ = 1 this isprecisely the extra term found in the approximation resuls in [16, 19, 17]. One can thus interpretthese results, saying that the approximate solutions actually converge to solutions of the rightequation with a different choice of reference rough path.

In the gradient case g = DG our solution coincides with the classical solution defined usingintegration by parts. To see this, it is sufficient to see that the nonlinearity can be rewritten as∫ 1

0ϕ(x) g

(u(s, x)

)dxu(s, x) = −

∫∂xϕ(x)G

(u(x)

)dx.

For simplicity, let us argue component-wise and show this formula for a function G that takes valuesin R, i.e, we have g = ∇G. We will assume that G is of class C3 with bounded derivatives up toorder 3. To simplify the notation we leave out the time dependence and write ϕi, ui for ϕ(xi), u(xi)etc.

Then we can write∫ 1

0ϕ(x) g

(u(x)

)dxu(x)

= lim∑i

ϕi

[∇G(ui) · δui,i+1 +

(∂α,βG(ui)

(u′i)αα′(

u′i)ββ′

Xα′β′

i,i+1

)]= lim

∑i

ϕi

[G(ui+1)−G(ui) +Qi

]. (3.13)

The sum involving the differences of G converges to −∫∂xϕGdx, so that it only remains to bound

Qi. Here the crucial ingredient is the fact that the Gaussian rough paths constructed by Friz andVictoir are geometric (see Remark 2.4). In particular we have for all x, y:

Sym(X(x, y)

)= δX(x, y)⊗ δX(x, y). (3.14)

ROUGH BURGERS 13

As D2G is a symmetric matrix the antisymmetric part of X does not contribute towards the secondterm in the second line of (3.13) and we can rewrite with Taylor formula

|Qi| ≤∣∣D3G

∣∣0|δui,i+1|3 +

∣∣D2G∣∣0|Ru(xi, xi+1)|2

≤ |G|C3

(|u|3Cα

(xi+1 − xi

)3α+ |Ru|22α

(xi+1 − xi

)4α). (3.15)

This shows that the sum involving Qi on the right hand side of (3.13) vanishes in the limit.This discussion shows in particular that in the gradient case g = DG the concept of weak solution

is independent of the reference rough path if we impose the additional assumption that it is geometrici.e. that it satisfies (3.14). The extra reaction term we obtained above in (3.12) is due to the fact thatthe modified rough path (X, X) is not geometric.

In the non-gradient case this is no longer true. Extra terms may appear even for geometric roughpaths. To see this, one can for example consider the two-dimensional example

X(t, x, y) = X(t, x, y) + (y − x)

(0 1−1 0

)(3.16)

and

g(u1, u2) =

(u2 −u1

u2 −u1

). (3.17)

Then a similar calculation to (3.12) shows that an extra reaction term

Tr

[(0 1−1 0

)θ(u)

(0 1−1 0

)θ(u)T

](3.18)

appears in each coordinate.We justify our concept of solution by showing that it is stable under smooth approximations. To

this end let ηε(x) = ε−1η(x/ε) be standard mollifiers as introduced in (4.2). For a fixed initial datau0 ∈ Cβ denote by uε the unique solution to the smoothened equation

duε =[∆uε + g(uε)∂xuε

]dt+ θ(uε) d

(W ∗ ηε

). (3.19)

Note that due to the spatial smoothening of the noise there is no ambiguity whatsoever in theinterpretation of (3.19) . We then get

Theorem 3.6. Fix 14 < α < β < 1

2 . Fix an initial data u0 ∈ Cβ and assume that g and θ are as inTheorem 3.5. Then we have for any δ > 0 and every T > 0

P[‖u− uε‖Cα/2,α[0,T ]×[0,1] ≥ δ

]→ 0 as ε ↓ 0. (3.20)

Recall that the parabolic Holder norm on ‖u‖Cα/2,α[0,T ]×[0,1] was defined in (3.2).

4. STOCHASTIC CONVOLUTIONS

This section deals with properties of stochastic convolutions. Fix a probability space (Ω,F ,P)with a right continuous, complete filtration (Ft). Let W (t) = (W 1(t), . . . ,Wn(t)), t ≥ 0 be acylindrical Wiener process on L2

([0, 1],Rn

). Then for any adapted L2

([0, 1],Rn×n

)valued process

θ define

Ψθ(t) =

∫ t

0S(t− s) θ(s) dW (s). (4.1)

Recall that S(t) = et∆ denotes the semigroup associated to the Laplacian with periodic boundaryconditions, which acts independently on all of the components. The Gaussian case θ = 1 will play aspecial role and we will denote it with X = Ψ1.


We will also treat the stochastic convolution for a smoothened version of the noise. To this endlet η : R → R be a smooth, non-negative, symmetric function with compact support and with∫η(x) dx = 1. Furthermore, for ε > 0

ηε(x) = ε−1η(xε

). (4.2)

Then denote by

Ψθε(t) =

∫ t

0S(t− s) θ(s) d

(W ∗ ηε

)(s), (4.3)

the solution to the stochastic heat equation with spatially smoothened noise. As above in the specialcase θ = 1 we will write Xε = Ψ1

ε .The main purpose of this section is to establish the following facts: In the Gaussian case θ = 1

for every t the stochastic convolution X(t, ·) = Ψ1 satisfies the criteria of Lemma 2.3 and thereforecan be lifted to a rough path (X(t, ·),X(t, ·)) in space direction. Furthermore, we will be able toconclude that (X(t, ·),X(t, ·)) is a continuous process taking values in Dα.

If θ 6= 1 is not deterministic, the process Ψθ is in general not Gaussian and Lemma 2.3 doesnot apply. But we will establish that as soon as θ has the right space-time regularity the stochasticconvolution Ψθ(t, ·) is controlled by X(t, ·) for every t. The essential ingredients are the estimatesin Lemma 4.2. Finally, we will show that all of these results are stable under approximation with thesmoothened version defined in (4.3).

Recall that for f ∈ L2[0, 1] we can write

S(t)f(x) =

∫ 1

0pt(x− y)f(y) dy, (4.4)

and that one has an explicit expression for the heat kernel pt either in terms of the Fourier decompo-sition

pt(x) =∑k∈Z

exp(− (2πk)2t

)exp

(i(2πk)(x)

), (4.5)

or from a reflection principle

pt(x) =∑k∈Z

pt(x− k), (4.6)

where pt(x) = 1(4πt)1/2 exp

(− x2/4t

)denotes the usual Gaussian heat kernel.

The following two lemmas about some integrals involving pt are the central part of the proof ofregularity for the stochastic convolutions. We first recall the following bounds:

Lemma 4.1. The following bounds hold:

(1) Spatial regularity: For x, y ∈ [0, 1] and t ∈ [0, T ]∫ t

0

∫ 1

0

(pt−s(x− z)− pt−s(y − z)

)2ds dz ≤ C

∣∣x− y∣∣. (4.7)

(2) Temporal regularity: For x ∈ [0, 1] and 0 ≤ s < t ≤ T∫ s

0

∫ 1

0

(pt−τ (x− z)− ps−τ (x− z)

)2dτ dz

+

∫ t

s

∫ 1

0

(pt−τ (x− z)

)2dτ dz ≤ C(t− s)1/2. (4.8)

ROUGH BURGERS 15

Proof. These bounds are essentially well-known and can be derived easily from the expression (4.5).To see (4.7) write∫ t

0

∫ 1

0

(pt−s(x− z)− pt−s(y − z)

)2ds dz

=∞∑k=1

2

(2πk)2

[1− exp

(− 2(2πk)2t

)]sin(πk(x− y)

)2

≤ C∞∑k=1

1

k2

(k2(x− y)2 ∧ 1

)≤ C

(x− y

). (4.9)

In order to see (4.8) write first∫ s

0

∫ 1

0

(pt−τ (x− z)− ps−τ (x− z)

)2dτ dz

=

∞∑k=1

1

(2πk)2

[1− exp

(−(2πk

)2(t− s)

)]2[1− exp

(2(2πk

)2s)]

≤ C

∞∑k=1

1

k2

[(2πk

)2(t− s) ∧ 1

]2≤ C(t− s)1/2, (4.10)

and ∫ t

s

∫ 1

0

(pt−τ (x− z)

)2dτ dz =

∞∑k=1

1(2πk

)2 [1− exp(

2(2πk

)2(t− s)

)]≤ C

∞∑k=1

1

k2

[k2(t− s) ∧ 1

]≤ C(t− s)1/2. (4.11)

Then combining (4.10) and (4.11) yields (4.8).

The following lemma is the essential step to prove that Ψθ is controlled by X:

Lemma 4.2. The following bounds hold:(1) Regularisation by a time dependent function: For all x, y ∈ [0, 1] and every t ∈ [0, T ]∫ t

0

∫ 1

0

(pt−s(y − z)− pt−s(x− z)

)2|s− t|α dz ds ≤ C|x− y|1+2α. (4.12)

(2) Regularisation by a space dependent function: For all x, y ∈ [0, 1] and every t ∈ [0, T ]∫ t

0

∫ 1

0

(pt−s(y − z)− pt−s(x− z)

)2|x− z|2α dz ds ≤ C|x− y|1+2α. (4.13)

Proof. Due the periodicity of pt we can assume that x = 0 and replace the spatial integral by anintegral over [−1

2 ,12 ]. Furthermore, without loss of generality we can assume that 0 ≤ y ≤ 1

4 . Dueto the explicit formula (4.6) one can see that for z ∈ [−3

4 ,34 ] one can write pt(z) = pt(z) + pRt (z)

where the remainder pRt is a smooth function with uniformly bounded derivatives of all orders. Thus,using (

pt−s(z − y)−pt−s(z))2

≤ 2(pt−s(z − y)− pt−s(z)

)2+ 2(pRt−s(z − y)− pRt−s(z)

)2,


and that ∫ t

0

∫ 1

0

(pRt−s(z − y)− pRt−s(z)

)2|s− t|α dz ds ≤ C|pR|C1 |y|2, (4.14)

as well as the analogous bound involving he term |x− y|2α, one sees that it is sufficient to prove(4.12) and (4.13) with pt replaced by the Gaussian heat kernel pt.

(i) Let us treat the integral involving |s − t|α first. To simplify notation we denote by I1 =∫ t0

∫ 12

− 12

(pt−s(z − y)− pt−s(z)

)2|s− t|α dz ds. A direct calculation shows

∫R

(pt−s(y − z)− pt−s(z)

)2dz =

1√2π

(t− s)−1/2

[1− exp

(− y2

8(t− s)

)]. (4.15)

So one gets:

I1 ≤1√2π

∫ t

0(t− s)α−1/2

[1− exp

(− y2

8(t− s)

)]ds

=1√2π

8−1/2−α|y|1+2α

∫ ∞y2

8t

τ−3/2−α[1− e−τ ]dτ. (4.16)

Here in the last step we have performed the change of variable τ = y2

8(t−s) . Note that the integralin the last line of (4.16) converges at 0 due to α < 1

2 . Thus we can conclude

I1 ≤1√2π

8−1/2−α|y|1+2α

∫ ∞0

τ−3/2−α[1− e−τ ]dτ ≤ C|y|1+2α. (4.17)

(ii) Let us now prove the bound (4.13). We write

I2 =

∫ t

0

∫ 1

0


)2|z|2α dz ds.

We decompose the integral into

I2 ≤∫ t

(t−|y|2)∧0

∫R


)2|z|2α dz ds (4.18)

+

∫ (t−|y|2)∧0

0

∫R


)2|z|2α dz ds

= I2,1 + I2,2.

For the first term we get

I2,1 ≤ 2

∫ t

(t−|y|2)∧0

∫R

[(pt−s(z − y)

)2+(pt−s(z)

)2]|z|2α dz ds

≤C∫ t

(t−|−y|2)∧0

∫R

(t− s)−1 exp

(− z2

2(t− s)

)(|z|2α + |y|2α

)dz ds

≤C∫ t

(t−|y|2)∧0(t− s)−1+1/2+α ds+ C

∫ t

(t−|y|2)∧0(t− s)−1+1/2|y|2α ds

≤C|y|1+2α. (4.19)

ROUGH BURGERS 17

In order to get an estimate on the second term we write(pt−s(z − y)− pt−s(z)

)2≤(|y| sup

ξ∈[z−y,z]

∣∣p′t−s(ξ)∣∣)2. (4.20)

We have for t ≥ y2

supξ∈[z−y,z]

∣∣p′t(ξ)∣∣ ≤ supξ∈[z−y,z]

|ξ|2t√

4πtexp

(− ξ2

4t

)≤ |z|+ |y|

2t√

4πtexp

(− z2

8t

)e1/4, (4.21)

where we have used the the identity (a+ b)2 ≥ 12a

2 − b2 as well as t ≥ y2. Thus we can write

I2,2 ≤ C|y|2∫ (t−y2)∧0

0

∫R

z2 + y2

(t− s)3exp

(− z2

4(t− s)

)|z|2α dz ds

≤ Cy2

∫ (t−y2)∧0

0

(t− s) + y2

(t− s)3(t− s)1/2+αds

≤ C|y|1+2α. (4.22)

This completes the proof.

It is a well known fact and an immediate consequence of Lemma 4.1 together with the Kolmogorovcriterion that (say for bounded θ) for any α < 1

2 the stochastic convolution Ψθ is almost surelyα-Holder continuous as a function of the space variable x for a fixed value of the time-variable t andalmost surely α/2-Holder continuous as function of the times variable t for a fixed value of x. Thefollowing proposition makes this statement more precise:

Proposition 4.3. For p ≥ 2 denote by

|||θ|||p,0 = E[

sup(t,x)|θ(t, x)|p

]1/p. (4.23)

For every α < 12 and every p > 12

1−2α set κ = 1−2α4 − 3

p . Then we have

E[∥∥Ψθ

∥∥pCκ([0,T ];Cα)

]≤ C|||θ|||pp,0. (4.24)

Remark 4.4. In particular using that Ψ(0, x) = 0 this proposition implies that

E[∥∥Ψθ

∥∥pC([0,T ];Cα)

]≤ CT κ|||θ|||pp,0. (4.25)

On the other hand, by interchanging the roles of κ and α, one obtains E[∥∥Ψθ

∥∥pC(α+κ)/2([0,T ];Cκ)

]≤

C|||θ|||pp,0 and in particular

E[∥∥Ψθ

∥∥pCα/2([0,T ];C0)

]≤ CT κ/2|||θ|||pp,0. (4.26)


Proof. Fix an 0 ≤ s < t ≤ T and x, y ∈ [0, 1]. For every β ∈ (0, 1/2) we can write using Holder’sinequality

E

[((Ψ(t, x)−Ψ(s, x)

)−(Ψ(t, y)−Ψ(s, y)

)|x− y|β

)p]

≤ E

[(∣∣Ψ(t, x)−Ψ(t, y)∣∣

|x− y|1/2+

∣∣Ψ(s, x)−Ψ(s, y)∣∣

|x− y|1/2

)p ]2β

E[(∣∣Ψ(t, x)−Ψ(s, x)

∣∣+∣∣Ψ(t, y)−Ψ(s, y)

∣∣)p]1−2β. (4.27)

To bound the first term on the right hand side of (4.27) we write using BDG-inequality ([21, Theorem3.28 p. 166])

E

[(∣∣Ψ(t, x)−Ψ(t, y)∣∣

|x− y|1/2

)p ]

≤ C

|x− y|p/2E

[(∫ t

0

∫ 1

0

(pt−s(z − x)− pt−s(z − y)

)2θ(s, z)2dsdz

)p/2]≤ C|||θ|||pp,0. (4.28)

Here in the last step we have used (4.7). A very similar calculation involving (4.8) shows that

E[∣∣Ψ(t, x)−Ψ(s, x)

∣∣p] ≤ C|||θ|||pp,0(t− s)p/4. (4.29)

So combining (4.27) - (4.29) we get

E

[((Ψ(t, x)−Ψ(s, x)

)−(Ψ(t, y)−Ψ(s, y)

)|x− y|β

)p]≤ C|||θ|||pp,0(t− s)p/4(1−2β). (4.30)

Now applying Lemma 2.11 to the function f = Ψθ(t, ·)−Ψθ(s, ·) one gets that for x 6= y andfor p > 2

β ∣∣f(x)− f(y)∣∣ ≤ CU1/p|x− y|β−2/p, (4.31)

where

U =

∫ 1

0

∫ 1

0

(|f(x)− f(y)||x− y|β

)pdx dy. (4.32)

Noting that (4.30) implies that E[U]≤ C|||θ|||pp,0(t− s)p/4 one can conclude that

E[|Ψ(t, ·)−Ψ(s, ·)|p

β− 2p

]≤ C|||θ|||pp,0(t− s)p(1−2β)/4. (4.33)

Using the identity |Ψ(t, ·)−Ψ(s, ·)|0 ≤ C(|Ψ(t, 0)−Ψ(s, 0)|0 + |Ψ(t, ·)−Ψ(s, ·)|α

)and using

(4.29) as well as (4.30) we get the same bound for the expectation of |Ψ(t, ·) − Ψ(s, ·)|0 so thatfinally we get

E[|Ψ(t, ·)−Ψ(s, ·)|p

Cβ− 2

p

]≤ C|||θ|||pp,0(t− s)p(1−2β)/4. (4.34)

Now setting β = α+ 2p , which is less than 1

2 by the assumtion on p, and using Lemma 2.11 oncemore in the time variable yields the desired result.

ROUGH BURGERS 19

In order to get stability of this regularity result under smooth approximations it is useful to imposea regularity assumption on θ. For any α < 1

2 and for p ≥ 2 denote by

‖θ‖p,α = E[

supx 6=y,s6=t

|θ(t, x)− θ(s, y)|p(|t− s|α/2 + |x− y|α

)p + supx,t|θ(t, x)|

]1/p

. (4.35)

Then we have the following result:

Proposition 4.5. Choose α and p as in Proposition 4.3. Then for any

γ < α(

1− 2α− 12

p

)(4.36)

there exists a constant C such that for any ε ∈ [0, 1]

E[∥∥Ψθ −Ψθ

ε

∥∥pC([0,T ];Cα)

]≤ C|||θ|||pp,αεpγ (4.37)

and

E[∥∥Ψθ −Ψθ

ε

∥∥pCα/2([0,T ];C)

]≤ C|||θ|||pp,αεpγ . (4.38)

Remark 4.6. Actually we only need the spatial Holder regularity of θ in the proof. We introducethe space-time parabolic regularity condition on θ as we will need it in the sequel.

Proof. We first establish the bound

E[∣∣Ψε(t, x)−Ψ(t, x)

∣∣p] ≤ Cεαp, (4.39)

where the constant C is uniform in x and t ≤ T .By BDG-inequality and the definitions of Ψθ

ε and Ψθ we get

E[∣∣Ψε(t, x)−Ψ(t, x)

∣∣p]≤ C E

[(∫ t

0

∫ 1

0

(∫ 1

0ηε(y − z)

(pt−s(x− y) θ(s, y)− pt−s(x− z) θ(s, z)

)dz

)2

dy ds

) p2].

(4.40)

The inner integral on the right hand side of (4.40) can be bounded by∫ 1

0ηε(y − z)

(pt−s(x− y) θ(s, y)− pt−s(x− z) θ(s, z)

)dz

≤ |θ(t)|α pt−s(x− y)

∫ 1

0ηε(y − z) |y − z|α dz

+ |θ(t)|0∫ 1

0ηε(y − z)|pt−s(x− y)− pt−s(x− z)|dz. (4.41)

For the first term on the right hand side of (4.41) we get

E[(∫ t

0

∫ 1

0

(|θ(t)|α pt−s(x− y)

∫ 1

0ηε(y − z) |y − z|α dz

)2dy ds

) p2]

≤ Cεαp E[(∫ t

0

∫ 1

0|θ(t)|2α p2

t−s(x− y) dy ds

) p2]

≤ C |||θ|||pp.αεαp. (4.42)


For the second term in the right hand side of (4.41) we devide the time integral into the integral overs ≥ t− ε2 and the integral over s ≤ t− ε2. For the first one we get using Young’s inequality

E[(∫ t

t−ε2

∫ 1

0

(∫ 1

0|θ(t)|0ηε(y − z)|pt−s(x− y)− pt−s(x− z)|dz

)2dy ds

) p2]

≤ C |||θ|||p,0(∫ t

t−ε2

∫ 1

0p2t−s(x− y) dy

) p2

≤ C |||θ|||p,0 εp2 . (4.43)

Using Young’s inequality again the term involving the integral over s ≤ t− ε2 can be bounded by

|||θ|||pp,0(∫ t−ε2

0

∫ 1

0

∫ 1

0ηε(y − z)|pt−s(x− y)− pt−s(x− z)|2dz dy

ds

) p2

. (4.44)

Similarly to the proof of Lemma 4.2 we decompose pt = pt + pRt into the Gaussian heat kernel anda smooth remainder. The term in (4.44) with pt replaced by pRt can be bounded by C |||θ|||pp,0 εp so itis sufficient to consider (4.44) with pt replaced by pt. But then we get from the scaling property ofpt

|||θ|||pp,0(∫ t−ε2

0

∫ 1

0

∫ 1

0ηε(y − z)|pt−s(x− y)− pt−s(x− z)|2dz dy

ds

) p2

≤ |||θ|||pp,0(∫ t−ε2

0

ε3

(t− s)2

∫R

∫Rη1(y − z) |p′1|0|y − z)|2dz dy

ds

) p2

≤ C |||θ|||pp,0εp2 . (4.45)

Thus combining (4.42), (4.43) and (4.45) we obtain the desired bound (4.39).The rest of the argument follows along the same lines as the proof of Proposition 4.3. Writing

fε(t, x) = Ψθ(t, s)−Ψθε(t, s) we get similarly to (4.27):

E

[((fε(t, x)− fε(s, x)

)−(fε(t, y)− fε(s, y)

)|x− y|β

)p]

≤ C supt,ε

E

[(∣∣fε(t, x)− fε(t, y)∣∣

|x− y|1/2

)p ]2β1

supx,ε

E[(∣∣fε(t, x)− fε(s, x)

∣∣)p]1−2β1−β2

supx,t

E[(∣∣fε(t, x)

∣∣)p]β2

. (4.46)

Noting that due to Young’s inequality, the bounds on the space-time regularity of Ψ (4.28) and (4.29)also hold for Ψε with a constant independent of ε. Using (4.39) we can bound the right hand side of(4.46) by

E

[((fε(t, x)− fε(s, x)

)−(fε(t, y)− fε(s, y)

)|x− y|β

)p]≤ C|||θ|||pp,α(t− s)

p(1−2β1−β2)4 ε

β2p2 .

Thus, as in the proof of Proposition 4.3, by applying the Garsia-Rodemich-Rumsey Lemma 2.11twice we get the desired result.

In the Gaussian case one can apply the results from [9] to obtain a canonical rough path versionof X . We have

ROUGH BURGERS 21

Proposition 4.7. For a fixed time t the stochastic convolution X(t, x) viewed as a process in xcan be lifted canonically to an α rough path, which we denote by

(X(t),X(t)

). Furthermore, the

process(X(t),X(t)

)is almost surely continuous and one has for all p

E[∥∥X∥∥p

ΩC2αT

]< ∞. (4.47)

Furthermore, for ε > 0 the Gaussian paths Xε are smooth and Xε can be defined using (2.5). Thenone has for ε ↓ 0

E[∥∥X−Xε

∥∥pΩC2α

T

]→ 0. (4.48)

Here∥∥X∥∥

ΩC2αT

= sup0≤t≤T∣∣X(t)

∣∣ΩC2α .

Proof. For a given t ∈ [0, T ] the covariance function of every component of X(t, ·) is given byE[X(t, x)X(t, y)] = Kt(x− y) + t where

Kt(x) =∑k≥1

1

(2πk)2

[1− exp(−2(2πk)2t))

]cos(2πkx

). (4.49)

The constant term t does not contribute to the two-dimensional 1-variation of the covariance. Totreat the second term note that

K1t (x) =

∑k≥1

1

(2πk)2cos(2πkx

)=

1

2|x| − 1

2x2 − 1

12, (4.50)

for x ∈ [−1, 1] and periodically extended outside of this interval. In particular, the two-dimensional1-variation of the term corresponding to K1

t is given as

|K1t (x− y)|1−var[x1, x2]× [y1, y2] =

∫ x2

x1

∫ y2

y1

(1 + δx=y

)dx dy. (4.51)

Noting that the second term

K2t (x) = −

∑k≥1

1

(2πk)2

[exp(−2(2πk)2t))

]cos(2πkx

)(4.52)

is given by the convolution (on the torus) of K1t with the heat kernel pt one can easily see that the

two-dimensional 1-variation of K satisfies condition (2.8) with a uniform constant in t. This showsthat for every t the process X(t, ·) can be lifted to a rough path.

To deal with the continuity note that by (4.29)

E[(X(t, x)−X(s, x)

)2] ≤ C|t− s|1/4, (4.53)

such that by in interpolation argument one gets

|KX(t)−X(s)|β−var ≤ C|t− s|κ (4.54)

for any α < β < 1 and a small θ > 0. Thus the second part of Lemma 2.3 and an application of theGarsia-Rodemich-Rumsey Lemma 2.11 yield the result. The bound (4.48) follows in the same waynoting that in the case θ = 1 the bound (4.39) reads

E[|Xε(t, x)−X(t, x)|p

]≤ Cε

p2 . (4.55)


As the processes Ψθ are not Gaussian for general θ we cannot draw the same conclusion to defineiterated integrals for Ψθ. The crucial observation is that as soon as θ possesses a certain regularitythe worst fluctuations are controlled by those of the Gaussian process X .

For K > 0 and for α ∈ (0, 1/2) we introduce the stopping time

τ‖X‖αK = inf

t ∈ [0, T ] : sup

x1 6=x20≤s1<s2≤t

∣∣X(s1, x1)−X(s2, x2)∣∣

|s1 − s2|α/2 + |x1 − x2|α> K

(4.56)

Note that by Proposition 4.3 for every α < 12 one has τ‖X‖αK > 0 and limK↑∞ τK = T almost

surely.

Proposition 4.8. Denote by

Rθ(t, x, y) = Ψθ(t, y)−Ψθ(t, x)− θ(t, x)(X(t, y)−X(t, x)

). (4.57)

For every

p >2(6α+ 2)

α(1− 2α)(4.58)

set

κ =α(1− 2α)

2(1 + 2α)− 6α+ 2

p(1 + 2α)> 0. (4.59)

Then one has

E[∥∥Rθ∥∥p

Cκ(

[0,τ‖X‖αK ];ΩC2α

)] ≤ C(1 +Kp

)‖θ‖pp,α. (4.60)

Remark 4.9. Similar to above (4.60) implies in particular that

E[∥∥Rθ∥∥p

C0(

[0,τ‖X‖αK ];ΩC2α

)] ≤ CT κ(1 +Kp

)‖θ‖pp,α. (4.61)

Proof. Similar to the proof of Proposition 4.3 we begin by noting that for 0 ≤ s < t ≤ T and forany β ∈ (0, 1

2 + α),

E[(

Rθ(t, x, y)−Rθ(s, x, y)

|x− y|β

)p]

≤ E

[(∣∣Rθ(t, x, y)∣∣− ∣∣Rθ(s, x, y)

∣∣|x− y|

12

+α

)p] 2β1+2α

E[∣∣Rθ(t, x, y)−Rθ(s, x, y)

∣∣p](1− 2β1+2α

).

(4.62)

Using the definition (4.57)of Rθ(t, x, y) one obtains

Rθ(t, x, y) =

∫ t

0

∫R

(pt−s(z − y)− pt−s(z − x)

)(θ(s, z)− θ(t, x)

)W (ds, dz). (4.63)

ROUGH BURGERS 23

So by BDG inequality we have

E[(Rθ(t, x, y)

)p]≤ C E

[(∫ t

0

∫R

(pt−s(y, z)− pt−s(x, z)

)2(θ(s, z)− θ(t, x)

)2dz ds

)p/2].

≤ C E

[sup

z 6=x,s 6=t

(θ(s, z)− θ(t, x)

|z|α + |s− t|α/2

)p](∫ t

0

∫R

(pt−s(z − y)− pt−s(z − x)

)2(|z − x|α + |s− t|α/2

)2dz ds

)p/2≤ C‖θ‖pp,α |x− y|(1+2α)p/2. (4.64)

Here we have used (4.12) and (4.13) in the last line.Similarly, one can write using (4.57)

E[∣∣Rθ(t, x, y)−Rθ(s, x, y)

∣∣p]≤ C E

[∣∣Ψθ(t, y)−Ψθ(s, y)∣∣p]+ C E

[∣∣Ψθ(s, x)−Ψθ(t, x)∣∣p]

+ C E[∣∣θ(t, x)− θ(s, x)

∣∣p∣∣X(t, y)−X(t, x)∣∣p]

+ C E[|θ(s, x)|p

∣∣X(t, y)−X(s, y)∣∣p]+ C E

[|θ(s, x)|p

∣∣X(s, x)−X(t, x)∣∣p]

= C(I1 + I2 + I3 + I4 + I5

). (4.65)

The terms I1 and I2 are bounded by C|||θ|||pp,0(t− s)p/4. To bound I3 we use the definition of τ‖X‖αK

E[∣∣θ(t, x)− θ(s, x)

∣∣p∣∣Ψ(t, y)−Ψ(t, x)∣∣p] ≤ Kp‖θ‖pp,α(t− s)αp/2. (4.66)

To bound I4 and I5 we write using the definition of τ‖X‖αK again

E[|θ(s, x)|p

∣∣X(t, y)−X(s, y)∣∣p] ≤ Kp|||θ|||pp,0(t− s)pα/2. (4.67)

Thus summarising (4.62) - (4.67) we get

E[(

Rθ(t, x, y)−Rθ(s, x, y)

|x− y|β

)p]≤ C|||θ|||pp,α

(Kp|t− s|pα/2

)(1− 2β1+2α

). (4.68)

To be able to apply Lemma 2.11 we need similar bounds for

NRθ(t, x, y, z) =(θ(t, y)− θ(t, x)

)(X(t, z)−X(t, y)

). (4.69)

Recall that the operator N was defined in (2.2). Similar to the calculation in (4.62) observe that

E

[sup

0≤x≤u<v<r≤y≤1

∣∣NRθ(t, u, v, r)−NRθ(s, u, v, r)∣∣p|x− y|βp

]

≤ C E[

sup

∣∣NRθ(t, u, v, r)∣∣p +∣∣NRθ(s, u, v, r)∣∣p

|x− y|2αp

] β2α

E[

sup∣∣NRθ(t, u, v, r)−NRθ(s, u, v, r)∣∣p]1− β

2α

. (4.70)


To bound the first expectation we calculate using the definition of τ‖X‖αK

E[

sup

∣∣NRθ(t, u, v, r)∣∣p|x− y|αp

]≤ E

[∣∣θ(t, ·)∣∣pα

∣∣X(t, ·)∣∣pα

]≤ CKp|||θ|||pp,α. (4.71)

Similarly to (4.65) - (4.67) one can see that

E[

sup∣∣NRθ(t, u, v, r)−NRθ(s, u, v, r)∣∣p] ≤ C|||θ|||pp,α|t− s|pα/2. (4.72)

Thus applying Lemma 2.11 one gets

E[∣∣Rθ(t, ·, ·)−Rθ(s, ·, ·)∣∣

β− 2p

]≤ C(1 +Kp

1 )|||θ|||pp,α|t− s|pα/2(1− 2β

1+2α). (4.73)

Applying Lemma 2.11 once more in time direction and setting β = 2α+ 2p one obtains the desired

bound (4.60).

To finish this section, we show that the bound (4.60) is stable under approximations as well. Tothis end denote by

τ‖X‖α,εK = inf

t : sup

x1 6=x20≤s1<s2≤t

|(Xε(s1, x1))−Xε(s2, x2)

∣∣|s1 − s2|α/2 + |x1 − x2|α

> K

. (4.74)

Furthermore, we define the ε-remainder as

Rθε(t, x, y) = Ψθε(t, y)−Ψθ

ε(t, x)− θ(t, x)(Xε(t, y)−Xε(t, x)

). (4.75)

Proposition 4.10. Assume that |θ|0 is a deterministic constant. Then for any p as in (4.58) and forany

γ <α(1− 2α)

1 + 2α− 12α+ 4

p(1 + 2α)(4.76)

there exists a constant C such that

E[∥∥Rθ −Rθε∥∥pC0

([0,T∧τ‖X‖αK ∧τ‖X‖α,εK ];ΩC2α

)] ≤ C(1 +Kp

)‖θ‖pp,αεγ . (4.77)

Proof. Due to the deterministic bound on |θ|0 we get using (4.39) as well as the definition of Rθε that

E[∣∣Rθ(t, x, y)−Rθε(t, x, y)

∣∣] ≤ C|||θ|||pp,αεαp. (4.78)

Then in the same way as in (4.46) we get for hε = Rθε −Rθ that

E[(

hε(t, x, y)− hε(s, x, y)

|x− y|β1

)p]≤ C|||θ|||pp,α(1 +Kp)

(|t− s|pα/2

)(1− 2β1+2α

)−β2εαβ2 . (4.79)

Here we have used again that by Young’s inequality the bounds on the space-time regularity of Rhold for Rε as well with constants that are uniform in ε. The bounds on Nhε are derived in thesame way as in the proof of (4.8). Then the proof is again finished by the Garsia-Rodemich-RumseyLemma 2.11.

ROUGH BURGERS 25

5. EXISTENCE OF SOLUTIONS

In this section we prove Theorem 3.5 and Theorem 3.6. As the argument is quite long andtechnical we give an outline of the proof.

We will construct mild solutions to the equation (1.1) i.e. solutions of the equation

u(t) = S(t)u0 +

∫ t

0S(t− s) θ(u(s)) dW (s) +

∫ t

0S(t− s) g

(u(s)

)∂xu(s) ds, (5.1)

where as above S(t) denotes the heat semigroup which is given by convolution with the heatkernel pt(x). Note that the two integral terms in (5.1) are of a very different nature: The firstone

∫ t0 S(t− s) θ(u(s)) dW (s) is a stochastic integral in time whereas the second term is a usual

Lebesgue integral in time and a rough integral against the heat kernel in space:

S(t− s) g(u(s))∂xu(s)(x) =

∫ 1

0pt−s(x− y) g(u(s, y)) dyu(s, y). (5.2)

These terms have different natural spaces for solving a fixed point argument. It thus seems advisableto separate the construction into two parts.

In the additive noise case [15] θ = 1 this can be done using the following trick. As it has beenobserved in several similar cases (see e.g. [7]) subtracting the solution X to the linearised equation(1.2) regularises the solution. Actually, we expect v = u − X − U to be a C1 function of thespace variable x. Here for technical convenience we have also removed the term involving the initialcondition by subtracting the solution U(t) = S(t)u0 to the linear heat equation with the given initialdata. Then for fixed

(X,X

)one can obtain v by solving the deterministic fixed point problem

v(t) =

∫ t

0S(t− s)

[g(v(s) +X(s) + U(s)

)∂x(v(s) + U(s)

)]ds

+

∫ t

0

∫ 1

0pt−s(· − y)g

(v(s, y) +X(s, y)

)dyX(s, y) ds (5.3)

in C([0, T ], C1[0, 1]

). Then by adding X and U to the v one can recover the solution u. The crucial

ingredient for this fixed point problem is the regularisation of the heat semigroup. Lemma 5.1 givesa way to express this property in the rough path context.

In the multiplicative noise case some extra care is necessary. For a general adapted process θ, thestochastic convolution Ψθ is not Gaussian and thus Lemma 2.3 can not be applied to get a canonicalrough path lift. But we have seen in Proposition 4.8 that it can be viewed as an X controlled roughpath.

Given a fixed controlled rough path valued process Ψ we can again perform a pathwise construc-tion to obtain vΨ for this particular rough path valued function in the same manner as before bysolving the fixed point equation (5.3) with X replaced by Ψ. Furthermore, vΨ depends continuouslyon Ψ (and on (X,X)). To be more precise, we will prove in Proposition 5.10 that the C([0, T ], C1)

and the C1/2([0, T ], C) norms depend continuously on the controlled rough path norm of Ψ.This can then finally be used to construct u as a solution to another fixed point problem in a space

of adapted stochastic processes possessing the right space-time regularity. We solve the fixed pointproblem

u 7→ U + vθ(u) + Ψθ(u)

with finite norm |||u|||p,α (recall that the space-time Holder norm ||| · |||p,α was defined in (4.35)). Hereit is crucial to assume regularity for the process u as the bounds that control the rough path norms ofΨθ(u) depend on the regularity of θ(u). This is where we need that space time norms of v dependcontinuously on the rough path norms of Ψθ.


In order to perform this fixed point argument we need to introduce several cutoffs. These willthen finally be removed in the last step of the proof by deriving some suitable a priori bounds.

The entire construction is continuous in all the data. Using this and the stability results derived inSection 4 we finally prove Theorem 3.6.

Fix functionsX(t, x) ∈ CαT = C([0, T ], Cα) and X(t, x, y))∈ ΩC2α

T = C([0, T ],ΩC2α). Weassume that for every fixed time t the pair

(X(t, ·),X(t, ·)

)is a rough path in space i.e. we assume

that (2.4) holds. This reference rough path will remain fixed throughout the first (deterministic) partof the construction. We will use the notation∥∥(X,X)

∥∥DαT

= sup0≤t≤T

∣∣(X(t, ·),X(t, ·))∣∣Dα . (5.4)

Of course, in the end X will be the Gaussian rough path constructed in Proposition 4.7.Furthermore, we fix a function Ψ ∈ CαT . We assume that for every t the function Ψ(t, ·) is an

X-controlled rough path i.e. that there are functions Ψ′ ∈ CαT and RΨ ∈ ΩC2αT such that for every t

(2.14) holds. We write‖Ψ‖CαX,T = sup

0≤t≤T|Ψ(t, ·)|Cα

X(t). (5.5)

Finally, we fix the initial data u0 ∈ Cβ for 13 < α < β < 1

2 . For t > 0 write U(t, ·) = S(t)u0.Note that by standard regularisation properties of the heat semigroup

|U(t, ·)|C2α ≤ Ctβ−2α

2 |u0|β and |U(t, ·)|C1 ≤ Ctβ−1

2 |u0|β. (5.6)

For v ∈ C1T = C([0, T ], C1) define the operator GT = GT,Ψ,u0 as

GT (v)(t, x) =

∫ t

0S(t− s)

[g(u(s)

)∂x(v(s) + U(s)

)]ds (x)

+

∫ t

0

∫ 1

0pt−s(x− y)g(u(y, s)) dyΨ(y, s) ds. (5.7)

Here to abreviate the notation we write u(t, x) = U(t, x) + Ψ(t, x) + v(t, x). The spatial integralin the second line of (5.7) is to be interpreted as a rough integral: For every s the path Ψ(s.·) isin CαX(s) and so is g

(u(s)

)= g

(v(s) + U(s) + Ψ(s)

)(see Lemma 5.5). Thus the integral can be

defined as in Lemma 2.6. Note that as the spatial integral is the limit of the approximations (2.18)they are in particular measurable in s and the temporal integral can be defined as a usual Lebesgueintegral.

The next lemma is a modification of Proposition 2.5 and Lemma 3.9 in [15]. It is the crucialingredient in proving the regularising properties of the convolution with the heat kernel if understoodin the rough path sense.

Lemma 5.1. Let (X,X) be an α rough path and Y,Z ∈ CαX . Furthermore, assume that f : R→ Ris a C1 function such that

|f |1,1 =∑k∈Z

supx∈[k,k+1]

|f(x)|+ |f ′(x)| (5.8)

is finite. Then for any λ0 > 0 there exists a C > 0 such that the following bound holds for everyλ ≥ λ0∣∣∣ ∫ 1

0f(λx)Y (x) dZ(x)

∣∣∣ ≤ C|f |1,1λ−α[|Y |0|Z|α + |RY |2α|Z|α + |X|α|Y ′|0|RZ |2α

+ |X|2α(|Y ′|0|Z ′|α + |Y ′|Cα |Z ′|0

)+ |X|2α|Y ′|0|Z|α

]. (5.9)

ROUGH BURGERS 27

Remark 5.2. In most of the sequel we will not need the detailed version (5.9) but instead work withthe simplified bound∣∣∣ ∫ 1

0f(λx)Y (x) dZ(x)

∣∣∣ ≤ C|f |1,1λ−α(1 + |(X,X)|Dα)|Y |CαX |Z|CαX . (5.10)

Note however, that to derive the a priori bounds to prove global existence we will need to use that in(5.9) we do not have any term involving the product

∣∣RY ∣∣2α∣∣RZ∣∣2α or the product |Y |α|Z|α .

Remark 5.3. Using the bound (2.23) in the proof one can see that the same scaling behaviour holdstrue for the difference of integrals with respect to different rough paths. Assume that (X, X) isanother α rough path and Y , Z ∈ Cα

X. Then one has∣∣∣ ∫ 1

0f(λx)Y (x) dZ(x)−

∫ 1

0f(λx)Y (x) dZ(x)

∣∣∣≤ C|f |1,1λ−α

[(|Y |CαX + |Y |Cα

X

)(|Z|CαX + |Z|Cα

X

)(|X − X|Cα + |X− X|2α

)+ C

(|Z|CαX + |Z|Cα

X

)(1 +

∣∣(X,X)|Dα +∣∣(X, X)|Dα)

·(|Y − Y |Cα + |Y ′ − Y ′|Cα + |RY −RY |2α

)+ C

(1 +

∣∣(X,X)|Dα +∣∣(X, X)|Dα)(|Y |CαX + |Y |Cα

X

)·(|Z − Z|Cα + |Z ′ − Z ′|Cα + |RZ −RZ |2α

)]. (5.11)

Remark 5.4. We will only apply Lemma 5.1 in the case when f(x) is the heat kernel pt or itsderivative. Actually using the expression (4.6) it is easy to see that for every t ∈ [0, 1] there arefunctions ft and gt such that supt∈[0,1] |ft|1,1 + |gt|1,1 < ∞ and such that pt(x) = 1√

tft(x/√t)

and ∂xpt(x) = 1t gt(x/√t)

. Then applying (5.10) one gets∣∣∣ ∫ 1

0∂xpt(x, y)Z(x)dX(x)

∣∣∣ ≤ Ctα/2−1(1 + |(X,X)|Dα

)|Y |CαX |Z|CαX . (5.12)

and a similar bound for p instead of ∂xpt with scaling tα/2−1 instead of tα/2−1/2 .

Proof. (of Lemma 5.1) Without loss of generality we can assume λ ∈ N. Then we can write∫ 1

0f(λx)Y (x) dZ(x) =

λ∑k=1

∫ 1

0f(x+ k)Yλ,k(x) dZλ,k(x), (5.13)

where Yλ.k(x) = Y((x + k)/λ

)and similarly Zλ.k(x) = Z

((x + k)/λ

). The integrals on the

right hand side have to be understood as rough integrals with respcect to the reference rough pathXλ,k(x) = X

((x−k)/λ

)and Xλ,k(x, y) = Xλ,k

((x−k)/λ, (y−k)/λ

). Then according to (2.19)

and (2.20) the integrals on the right hand side of (5.13) are given as∫ 1

0f(x+ k)Yλ,k(x) dZλ,k(x)

= f(k)Y

(k

λ

)δZ

(k

λ,k + 1

λ

)+ f(k)Y ′

(k

λ

)X

(k

λ,k + 1

λ

)Z ′(k

λ

)T+Qλ,k, (5.14)


where

|Qλ,k| ≤ Cαkλ−3α

[|Y |0|Z|α + |RY |2α|Z|α + |X|α|Y ′|0|RZ |2α+

|X|2α(|Y ′|0|Z ′|α + |Y ′|Cα |Z ′|0

)+ |X|2α|Y ′|0|Z|α

]. (5.15)

Here we have set αk = supx∈[k,k+1] |f(x)|+ |f ′(x)|. The first two terms on the right hand side of(5.14) can be bounded by

αk

(λ−α|Y |0|Z|α + λ−2α|Y ′|0|Z ′|0|X|2α

). (5.16)

Thus summing over k and recalling we obtain (5.9).

We need the following property:

Lemma 5.5. Let (X,X) ∈ Dα be a rough path and Ψ ∈ CαX . Furthermore, assume that w is a C2α

path and g is a C3 function with bounded derivatives up to order 3. Then

Y = g(Ψ + w) (5.17)

is a controlled rough path with derivative process Y ′ = Dg(Ψ + w)Ψ′. Furthermore, we have thefollowing bounds

|Y |Cα ≤ |g|0 + |Dg|0(|Ψ|α + |w|α

)|Y ′|Cα ≤ |Dg|0|Ψ′|Cα + |D2g|0|Ψ|0

(|Ψ|α + |w|α

)|RY |2α ≤ |Dg|0|w|2α + |D2g|0|Ψ|2α + |Dg|0|RΨ|2α. (5.18)

Furthermore let If (X, X) be another rough path, Ψ ∈ CαX

, and w ∈ C2α and write Y = g(Ψ + w).Then we have the following bounds

|Y − Y |Cα ≤ |g|C2

(1 + |Ψ|α + |Ψ|α + |w|α + |w|α

)(|Ψ− Ψ|Cα + |w − w|Cα

)|Y ′ − Y ′|Cα ≤C|g|C3

(1 + |Ψ|Cα + |Ψ|Cα + |w|Cα + |w|Cα

)[(|Ψ′ − Ψ|Cα

)+(|Ψ′|Cα + |Ψ|Cα

)(|Ψ− Ψ|Cα + |w − w|Cα

)]|RY −RY |2α ≤C|g|C3

[|w − w|2α +

∣∣RΨ −RΨ

∣∣2α

(5.19)

+(

1 + |Ψ|2Cα + |Ψ|2Cα + |w|C2α + |w|C2α

) (|Ψ− Ψ|Cα + |w − w|Cα

)].

Remark 5.6. It will sometimes be convenient to work with the following simplified version of(5.18)

|Y |CαX ≤ C|g|C2

(1 + |Ψ|2CαX

)(1 + |w|2α

). (5.20)

In the first fixed point argument we will use the following simplified version of (5.19) for the casewhen X = X and Ψ = Ψ:

|Y − Y |CαX ≤ C|g|C3

(1 + |Ψ|2CαX

)(1 + |w|C2α

)|w − w|C2α . (5.21)

Proof. We can write

δY (x, y) =

∫ 1

0Dg(Ψ(x) + w(x) + λ δΨ(x, y)

)δΨ(x, y) dλ

+

∫ 1

0Dg(Ψ(y) + w(x) + λ δw(x, y)

)δw(x, y) dλ. (5.22)

ROUGH BURGERS 29

The second integral can be bounded by |Dg|0|w|2α|x − y|2α. The first integral in (5.22) can berewritten as∫ 1

0

(Dg(Ψ(x) + w(x) + λ δΨ(x, y)

)−Dg

(Ψ(x) + w(x)

))δΨ(x, y) dλ

+Dg(Ψ(x) + w(x)

)δΨ(x, y). (5.23)

The integral in (5.23) can be bounded by |D2g|0|Ψ|2α|x− y|2α. Rewriting the term in the secondline as

Dg(Ψ + w

)δΨ = Dg

(Ψ + w

)(Ψ′δX +RΨ

)(5.24)

finishes the proof of (5.18).Let us now derive (5.19). To get a bound on the α -Holder semimorm of Y − Y write

δY (x, y)− δY (x, y)

= δ

∫ 1

0Dg(λ(Ψ + w) + (1− λ)(Ψ + w)

)(Ψ− Ψ + w − w

)dλ (x, y)

≤ |x− y|α∫ 1

0

∣∣∣Dg(λ(Ψ + w) + (1− λ)(Ψ + w))∣∣∣α

(∣∣Ψ− Ψ∣∣0

+∣∣w − w∣∣

0

)∣∣Dg(λ(Ψ + w) + (1− λ)(Ψ + w)

)∣∣0

(∣∣Ψ− Ψ∣∣∣α

+∣∣w − w∣∣

α

)dλ . (5.25)

This together with the bound |Y − Y | ≤ |g|C1

∣∣Ψ− Ψ∣∣0

+∣∣w− w∣∣

0yields the first bound in (5.19).

Note in particular, that Ψ and W appear quadratically in this estimate.The bound on

Y ′ − Y ′ = Dg(w + Ψ)Ψ′ −Dg(w + Ψ)Ψ′ (5.26)

is obtained in the same way. To treat the remainder RY observe that (5.22) - (5.24) show that

RY (x, y) =(g(Ψ(y) + w(y)

)− g(Ψ(y) + w(x)

))+Dg

(Ψ(x) + w(x)

)RΨ(x, y)∫ 1

0Dg(λ(Ψ(x) + w(x)) + (1− λ)(Ψ(y) + w(y)

)dλ δΨ(x, y), (5.27)

and similarly for RY . The bounds on the individual terms of RY (x, y)−RY (x, y) are obtained inan elementary way using similar integration as in (5.25) and we therefore omit the details.

Finally, we will need a version of Gronwall’s Lemma (See [27, Lemma 7.6] for a similarcalculation):

Lemma 5.7. Fix α, β ≥ 0 with α + β < 1 and a, b > 0. Let x : [0,∞) → [0,∞) be continuousand suppose that for every t ≥ 0

x(t) ≤ a+ b

∫ t

0(t− s)αs−βx(s)ds. (5.28)

Then there exists a constant cα,β that only depends on α, β such that for all t

x(t) ≤ a cα,β exp(cα,β b

11−β−α t

). (5.29)

Proof. Define inductively the sequences

A0(t) = 1 and R0(t) = x(t),

An+1(t) = b

∫ t

0(t− s)αs−βAn(s) ds Rn+1(t) = b

∫ t

0(t− s)αs−βRn(s) ds. (5.30)


Then one can show by induction that

An(t) =(b t1−β−α Γ(1− α)

)n n∏i=1

Γ(i− (i− 1)α− iβ

)Γ(i+ 1− iα− iβ

) . (5.31)

One can furthermore show by another induction using (5.28) that for any N

xt ≤ aN∑n=0

An(t) +RN+1(t) (5.32)

and that for all 0 ≤ t ≤ T

Rn(t) ≤(

sup0≤s≤T

xs

)An(t). (5.33)

In order to see (5.29) it thus remains only to bound the sum over the An. To this end note that theproduct in (5.31) is almost telescoping as due to the monotonicity of the Gamma function on [1,∞)

for i ≥ N0 =⌈

β1−α−β

⌉Γ(i+ 1− iα− iβ − β


) ≤ 1. (5.34)

For i < N0 we bound the same quotient by

Γ(i+ 1− iα− iβ − β


) ≤ c0 = supt∈[2−α−2β]

Γ(t), (5.35)

so that finally we get

βn(t) ≤(b t1−β−α Γ(1− α)

)nΓ(1− β)cN0

0

1

Γ(n+ 1− nα− nβ

) . (5.36)

Then the claim follows, as for all z ≥ 0 and γ > 0,

1 +∞∑n=1

zn

Γ(nγ + 1

) ≤ C(γ) e2z1/β. (5.37)

Actually, to see (5.37) one can assume without loss of generality that z ≥ 1 and calculate

1 +

∞∑n=1

zn

Γ(nγ + 1

) ≤ 1 +

∞∑k=0

( ∑n : bnγc=k

(z1/γ

)γnk!

)

≤ 1 +

∞∑k=0

⌈1

γ

⌉(z

(z1/γ

)kk!

)≤C(γ) z ez

1/γ ≤ C(γ) e2z1/γ. (5.38)

Setting γ = 1− β − α this finishes the argument.

Now we have all the ingredients to treat the operator GT defined in (5.7). For a fixed K > 0.Define

BK =v ∈ C1

T : ‖v‖C1T≤ K

, (5.39)

where we have set‖v‖C1

T= sup

0≤t≤T|v(t)|C1 . (5.40)

ROUGH BURGERS 31

Proposition 5.8. Let (X,X) and Ψ be as described above and fix the constant K > 0. Then thereexists a T = T (K, ‖(X,X)‖CαX,T , ‖Ψ‖CαX,T ) > 0 such that the mapping GΨ,T is a contraction onBK . In particular, there exists a unique fixed point which we will denote by vΨ.

Remark 5.9. The proof is very similar to the proof of Theorem 4.7 in [15]. We can not apply thisresult directly as here the integrator Ψ is an X controlled rough path instead of a rough path byitself. Of course, Ψ can be viewed as a rough path by itself, by defining the iterated integral as arough integral, but this would lead to problems in the next step of the construction as the mappingΨ 7→

∫δΨdΨ maps a stochastic Lp space continuously into a Lp/2 but not into itself, and it is not

clear how to define a suitable stopping time to avoid this problem.

Proof. We write

GT (v)(t, x) =

∫ t

0S(t− s)

[g(u(s)

)∂x(v(s) + U(s)

)]ds (x)

+

∫ t

0

∫ 1

0pt−s(x− y)g(u(y, s)) dyΨ(y, s) ds

=G(1)T v(t, x) +G

(2)T v(t, x). (5.41)

Let us treat the operator G(1)T first. Using the fact that the operator S(t) is bounded by Ct−1/2

from L∞ to C1 one gets

∣∣G(1)T v(t, ·)

∣∣C1 ≤

∫ t

0

∣∣S(t− s)∣∣L∞→C1

∣∣g∣∣0

(∣∣v(s)∣∣C1 +

∣∣U(s)∣∣C1

)ds

≤C|g|0(t1/2‖v‖C1

T+ t

β2 |u0|β

), (5.42)

such that for T small enough G(1)T maps BK into BK/2. Regarding the modulus of continuity of

G(1)T we write for v, v in BK

∣∣G(1)T v(t)−G(1)

T v(t)∣∣C1 ≤

∣∣∣∣ ∫ t

0S(t− s)

[(g(u(s)

)− g(u(s)

))∂y(v(s) + U(s)

)]ds

∣∣∣∣C1

+∣∣∣ ∫ t

0S(t− s)

[(g(v(s) + Ψ(s)

))∂y(v(s)− v(s)

)]ds∣∣∣C1,

such that

∣∣G(1)T v(t)−G(1)

T v(t)∣∣C1 ≤ C|g|C1

(t1/2(K + 1) + t

β2 |u0|β

)‖v − v‖C1

T.

For T small enough the last expression can be bounded by 13‖v − v‖C1

T.


Let us now treat the operator G(2)T . Using Lemma 5.1 (see also Remark 5.4) and then Lemma 5.5

as well as (5.20) we get∣∣∂xG(2)T v(t, x)

∣∣ =∣∣∣ ∫ t

0

∫ 1

0∂xpt−s(x− y)g

(u(s, y)

)dyΨ(s, y) ds

∣∣∣≤C

∫ t

0(t− s)

α2−1(1 + |(X(s),X(s)|Dα

) ∣∣g(u(s))∣∣CαX(s)

∣∣Ψ(s)∣∣CαX(s)

ds

≤C(1 + ‖(X,X‖DαT

)|g|C2∫ t

0(t− s)

α2−1(

1 +∣∣Ψ(s)

∣∣2CαX(s)

+K + |u0|β sβ−2α

2

)∣∣Ψ(s)∣∣CαX(s)

ds

≤C(1 + ‖(X,X‖DαT

)|g|C2

[tα2

(1 +K +

∥∥Ψ(s)∥∥3

CαX

)+ t

β−α2 |u0|β

∥∥Ψ∥∥CαX

].

A very similar calculation for the heat kernel without derivative using 5.1 (see also Remark 5.4)gives ∣∣G(2)

T v(t, x)∣∣ ≤ C

(1 + ‖(X,X‖DαT

)|g|C2(

1 +∣∣Ψ(s)

∣∣2CαX,T

)[tα+1

2

(1 +K

)+ t

β−α+12 |u0|β

]∥∥Ψ∥∥CαX. (5.43)

So we can conclude that for T small enough G(2)T maps BK into BK/2 as well.

To treat the modulus of continuity of G(2)T write

∂xG(2)T v(t, x)− ∂xG(2)

T v(t, x)

=

∫ t

0

∫ 1

0∂xpt−s(x, y)

(g(u(s, y)

)− g(u(s, y)

))dyΨ(s, y) ds. (5.44)

Recall that u(s) = v(s) + U(s) + Ψ(s). Using Lemma 5.5 we see that for every s the functiong(u(s)

)− g(u(s)

)is an X(s) controlled rough path. Thus applying (5.12) once more and then

using (5.21) we get∣∣∂xG(2)T v(t, x)− ∂xG(2)

T v(t, x)∣∣

≤C∫ t

0(t− s)

α2−1(1 + |(X(s),X(s)|Dα

) ∣∣g(u(s))− g(u(s)

)∣∣CαX(s)

∣∣Ψ(s)∣∣CαX(s)

ds

≤C(1 + ‖(X,X‖DαT

)|g|C3

(1 +

∣∣Ψ∣∣3CαX,T )∫ t

0(t− s)

α2−1(1 +K + |u0|β s

β−2α2

)|v(s)− v(s)|C1ds

≤C(1 + ‖(X,X‖DαT

)|g|C3

(1 +

∣∣Ψ∣∣3CαX,T )[tα2

(1 +K

)+ t

β−α2 |u0|β

]‖v(s)− v(s)‖C1

T. (5.45)

Repeating the same calculation yields a similar bound for∣∣G(2)

T v(t, x)−G(2)T v(t, x)

∣∣ with the same

exponents of t as in (5.43) . Thus for T small enough we can also bound∥∥G(2)

T v −G(2)T v∥∥C1T

by13‖v − v‖C1

T. This finishes the proof.

We now discuss the continuous dependence of the fixed point vΨ on the data. To this end let(X,X) ∈ DαT , Ψ ∈ CαX,T and the initial condition u0 be as above and let (X, X) ∈ DαT , Ψ ∈ Cα

X,T,

ROUGH BURGERS 33

u0 be another set of data. Assume that all the norms ‖(X,X)‖DαT , ‖Ψ‖CαX and |u0|β as well as thecorresponding norms for X , Ψ, and u are bounded by the constant K which defines the size of theball in which we solve the fixed point problem (see (5.39)). Note that by changing the existencetime of the fixed point argument this can always be achieved. In particular, the existence time Tonly depends on K and will be fixed for the next proposition

Proposition 5.10. Denote by v = vX,Ψ,u0 and by v = vX,Ψ,u0 the fixed points constructed inProposition 5.8 with the corresponding data. Then we have the following bounds

‖v − v‖C1T≤ C∆Ψ,Ψ (5.46)

and

‖v − v‖C1/2([0,T ],C0) ≤ C∆Ψ,Ψ, (5.47)

where we use ∆Ψ,Ψ as abbreviation for

∆Ψ,Ψ =

[‖X − X‖CαT + ‖X− X‖ΩC2α

T+ ‖Ψ− Ψ‖CαT + ‖Ψ′ − Ψ′‖CαT

+ ‖RΨ −RΨ‖ΩC2αT

+ |u0 − u0|β]. (5.48)

The constant C here depends on T,K, and |g|C2 .

Remark 5.11. Performing the same proof with a single set of data X,Ψ, u0 shows that one also hasthe bounds

‖v‖C1T≤ C

[‖(X,X)‖DαT + ‖Ψ‖CαX,T + |u0|β

](5.49)

and

‖v‖C1/2([0,T ],C0) ≤ C

[‖(X,X)‖DαT + ‖Ψ‖CαX,T + |u0|β

]. (5.50)

Proof. Let us show the bound (5.46) for the spatial regularity first. By the definition of v and v wecan write

v − v =(G(1)v − G(1)v

)+(G(2)v − G(2)v

). (5.51)

The operators G(i) and G(i) are the same as those defined as in (5.41) with respect to the respectivedata X,Ψ, u0 and X , Ψ, u0. To simplify the notation we omit the dependence on T .

As above we use that the heat semigroup S(t) is bounded by Ct−1/2 as operator from L∞ to C1

we get for the first term∣∣G(1)v(t)− G(1)v(t)∣∣C1 ≤ C(1 +K)|g|C1

∫ t

0(t− s)−1/2s

β−12

·(∣∣v(s)− v(s)

∣∣C1 +

∣∣Ψ(s)− Ψ(s)∣∣0

+ |u0 − u0|β)

ds. (5.52)

For the second term in (5.51) a similar calculation to (5.45) using the scaling bound for differentreference rough paths (5.11) shows∣∣G(2)v(t)− G(2)v(t)

∣∣C1 ≤ C |g|C3

∫ t

0(t− s)

α2−1(

1 +K4 +K3sβ−2α

2

)ds∆Ψ,Ψ

+ C |g|C3

∫ t

0(t− s)

α2−1(

1 +K4 +K3sβ−2α

2

)|v(s)− v(s)|C1ds. (5.53)


Now applying the Gronwall Lemma 5.7 to (5.52) and (5.53) yields (5.46).To treat the time regularity we write similarly to (5.51)(

v(t)− v(t))−(v(s)− v(s)

)=(Gv(t)− Gv(t)

)−(Gv(s)− Gv(s)

). (5.54)

We again separate G into G(1) and G(2) and get(G(1)v(t)− G(1)v(t)

)−(G(1)v(s)− G(1)v(s)

)=(S(t− s)− 1

)∫ s

0S(s− τ)

[g(u(τ)

)∂x(v(τ) + U(τ)

)− g(u(τ)

)∂x(vΨ(τ) + U(τ)

)]dτ

+

∫ t

sS(t− u)

[g(u(τ)

)∂x(v(τ) + U(τ)

)− g(u(τ)

)∂x(v(τ) + U(τ)

)]dτ. (5.55)

Note that here we write u = v + Ψ + U . The first term on the right hand side of (5.55) can bebounded using (5.46)

C∥∥S(t− s)− 1

∥∥C1→C0 |G(1)v(s)− G(1)v(s)|C1 ≤ C(t− s)

12 ∆Ψ,Ψ,

because∥∥S(t − s) − 1

∥∥C1→C0 ≤ C(t − s)1/2. The second term on the right hand side of (5.55)

can be bounded by∫ t

s

(|g|0(∣∣v(τ)− v(τ)

∣∣C1 + τ

β−12 |u0 − u0|β

)+(1 + τ

β−12)K|Dg|0

(∣∣v(τ)− v(τ)∣∣0

+∣∣Ψ(τ)− Ψ(τ)

∣∣0

+ |u0 − u0|β)

dτ

≤ C(t− s)β+1

2 ∆Ψ,Ψ,

where we use the fact that S(t− u) is a contraction from C0 into itself.For the term involving G(2) we write(G(2)v(t, x)− G(2)v(t, x)

)−(G(2)v(s, x)− G(2)v(s, x)

)=

∫ s

0

(∫ 1

0

(pt−τ (x− y)− ps−τ (x− y)

)g(u(τ, y)) dyΨ(τ, y)

−∫ 1

0

(pt−τ (x− y)− ps−τ (x− y)

)g(u(τ, y)

)dyΨ(τ, y)

)dτ (5.56)

+

∫ t

s

(∫ 1

0pt−τ (x− y)g

(u(τ, y)

)dyΨ(τ, y)−

∫ 1

0pt−τ (x− y) g

(v(y, τ)

)dyΨ(τ, y)

)dτ.

The second summand in (5.56) can be bounded as above (see e.g. (5.45), (5.53)) by:

C|g|C3(t− s)β−α+1

2 ∆Ψ,Ψ. (5.57)

In order to treat the first summand note that by the semigroup property

pt−τ (x− y) =

∫ 1

0pt−s(x− z)ps−u(z − y) dz. (5.58)

ROUGH BURGERS 35

Thus in the first integral in (5.56) we can rewrite using Lemma 2.10, the Fubini-Theorem for roughintegrals: ∫ 1

0pt−u(x− y) g

(u(τ, y)

)dyΨ(u, y) du

=

∫ 1

0

(∫ 1

0pt−s(x− z) ps−τ (z − y) dz

)g(u(τ, y)

)dyΨ(τ, y)

= S(t− s)(∫ 1

0ps−τ (· − y) g

(u(τ, y)

)dyΨ(τ, y)

)(x), (5.59)

and similarly for the integral involving Ψ. Thus the first difference in (5.56) can be bounded by∣∣S(t− s)− 1∣∣C1→C0

∣∣∣G(2)v(s)− G(2)v(s)∣∣∣C1≤ C(t− s)1/2∆Ψ,Ψ. (5.60)

Here we have used (5.46) again. This finishes the proof.

Now we are ready to construct local solutions to (1.1). To this end we introduce the followingfamilies of stopping times. For K1,K2,K3 > 0 and for any adapted processes Ψ, R taking valuesin Cα resp. ΩCα denote by

τXK1= inf

t ∈ [0, T ] : sup

x1 6=x20≤s1<s2≤t

∣∣X(s1, x1)−X(s2, x2)∣∣

|s1 − s2|α/2 + |x1 − x2|α+ |X(t)|2α > K1

τ|Ψ|CαK2

= inf

t ∈ [0, T ] : |Ψ(t)|Cα > K2

τ|R|2αK3

= inf

t ∈ [0, T ] : |R(t)|2α > K3

. (5.61)

Proposition 5.12. For any initial data u0 ∈ Cβ there exists a T ∗ > 0, an adapted process u∗(t, x)for (t, x) ∈ [0, T ∗] × [0, 1] and a stopping time σ3

K1,K2,K3such that up to T ∗ ∧ σ3

K1,K2,K3the

process u∗ satisfies (3.3). Furthermore, the time T ∗ only depends on |u0|β,K1,K2,K3 and thestopping time σ3

K1,K2,K3is given by

σ3K1,K2,K3

= τXK1∧ τ |Ψ

θ(u)|CαK2

∧ τ |Rθ(u)|2α

K3. (5.62)

Proof. We construct u by another fixed point argument, this time in the space of adapted stochasticprocesses on a time interval [0, T ], such that ||| · |||p,α is finite for suitably chosen p. (Recall that||| · |||p,α was defined in (4.35)). Let us denote the space of all these processes by Aα,p. Furthermore,denote by Aα,pK4

the set of processes u ∈ Aα,p with |||u|||p,α ≤ K4.We first need to introduce a cutoff function: Note that by definition of the stopping times for

t ≤ σ2K1,K2

and for u ∈ Aα,pK4one has ‖Ψθ(u)‖CαX,T ≤ C|θ|C1K4 + K2 + K3 = K5. Then for or

K5 > 1 let χK5 be a decreasing, non-negative C1 function which is constantly equal to one on(−∞,K5) and such that |χ′K5

|0 ≤ 1 and for all x ≥ K5

xχK5(x) ≤ 2K5. (5.63)

Now for u ∈ Aα,p define the operator N as

Nu(t) = S(tK1)u0 + Ψθ(u)(tK1) + vΨθ(u)K2 (tK1), (5.64)

where we write tK1 = t ∧ τXK1.


Here the Ψθ(u) is the stochastic convolution defined in (4.1) for the adapted process θ(u(t, x)

).

By Ψθ(u)K2

we denote the same stochastic convolution cut off at K2, i.e.

Ψθ(u)K2

(t) = Ψθ(u)(t)χK2

(∣∣Ψθ(u)(t)∣∣CαX(t)

). (5.65)

Finally, by vΨθ(u)K2 we denote the fixed point constructed in Proposition 5.8. Note that due to the

definition of χK2 clearly ‖Ψθ(u)K2‖CαX,T ≤ 2K2. Furthermore, due to the definition of the stopping

time τXK1all relevant norms of the reference rough path (X,X) are bounded by K1 so that this fixed

point is defined up to a final time T (K1,K5) ∧ τXK1that does not depend on u.

For T small enough and for K4 big enough (depending on |u0|β) the operator N maps Aα,pK4

into itself. Indeed, the deterministic part S(t)u0 has the right regularity due to the assumptionu0 ∈ Cβ and standard properties of the heat semigroup. Due to the boundedness of θ we can applyProposition 4.3 (see also Remark 4.4) to see that for p > 12

1−2α and for T small enough the stochasticconvolution Ψθ(u) also takes values in Aα,pK4

. Finally, as clearly |||θ(u)|||p,α ≤ |θ|C1

(1 + |||u|||p,α

)we

can apply Proposition 4.8 to conclude that for p satisfiying (4.58) Ψθ(u) is in fact a controlled roughpath with

E[

sup0≤t≤T∧τXK1

|RΨθ(u) |p2α]≤ C(1 +Kp

1 )T κ |θ|C1

(1 +K4

), (5.66)

for any κ satisfying (4.59). So Proposition 5.10 (see also Remark 5.11) implies that also vΨθ(u)K

takes values in Aα,pK4for T small enough. Note that K4 can be chosen depending only on |u0|β and

K1,K2,K3.Let us show that N is indeed a contraction. To this end for u, u calculate

|||Nu−N u|||p,α ≤ |||Ψθ(u) −Ψθ(u)|||p,α + |||vΨθ(u)K − vΨ

θ(u)K |||p,α. (5.67)

To deal with the stochastic convolutions note that by Proposition 4.3 and by Remark 4.4 for anyp > 12

1−2α there exists a κ > 0 such that

|||Ψθ(u)(t)−Ψθ(u)(t)|||p,α ≤CT κ/2|||θ(u)− θ(u)|||p,0≤CT κ/2|θ|C1 |||u− u|||p,0. (5.68)

So for T small enough this can be bounded by 13 |||u− u|||p,α.

A calculation similar to the proof of Lemma 5.5 shows that

|||θ(u)− θ(u)|||p,α ≤C(1 +K4)|θ|C2 |||u− u|||p,α. (5.69)

Note that in this bound the factor K4 appears and we do not expect a similar bound independent ofK4 to hold . It is the only place where we use that N is defined on a ball.

Therefore, using Proposition 4.8 we can see that for p even bigger (satisfying (4.58)) we canchose an even smaller κ (as in (4.59)) such that

E[∥∥Rθ(u) −Rθ(u)

∥∥pC(

[0,τXK1];ΩC2α

)] ≤ CT κ/2|||u− u|||pp,α. (5.70)

Now using (5.68) - (5.70) and the continuous dependence of the fixed point vΨ on the controlledrough path Ψ, Proposition 5.10, we get

|||vΨθ(u)K − vΨ

θ(u)K |||p,α ≤ CT κ/2|||u− u|||pp,α. (5.71)

ROUGH BURGERS 37

Choosing a T ∗ small enough this quantity can also be bounded by 13 |||u− u|||

pp,α. This shows that for

this T ∗ and for u, u ∈ Aα,pK4indeed |||Nu−N u|||p,α ≤ 2

3 |||u− u|||pp,α and in particular there exists a

unique fixed point u∗.To confirm that u∗ is indeed a mild solution up to T ∗ ∧ σ3

K1,K2,K3note that for t ≤ σ3

K1,K2,K3

the fixed point equation reads

u(t, x) =S(t)u0 + Ψθ(u)(t) + vΨθ(u)(t)

=S(t)u0 + Ψθ(u)(t) +GΨ(u)vΨθ(u)

(t), (5.72)

which is precisely the definition of a mild solution.

Now we are ready to finish the construction of global solutions to (1.1). We now show globalexistence as well as uniqueness:

Proof. (of Theorem 3.5) We will show global existence first. Fix a T > 0. We will show that wecan construct solutions up to time T . According to Proposition 5.12 for fixed K1,K2,K3 and anyinitial data u0 ∈ Cβ we can construct a process u∗ up to time T ∗ which then is a solution up tosome stopping time T ∗ ∧σ3

K1,K2,K3. By taking u∗(T ) as new initial condition and then iterating this

procedure (for fixed Ki) one can extend this process up to T ∧ T ∗∗ where T ∗∗ is the first blowuptime of |u∗|β .

A priori one should be careful at this point. The bounds used in construction of local solutionhave been derived using that the reference rough path X starts at 0. So in this way we get solutionswith a reference rough path that is restarted at T ∗. But the discussion after Definition 3.1 shows thatthis does not matter.

Using the definition of the fixed point u∗ we get for any t ≤ T that

|u∗(t)|β ≤ |u0|β + |Ψ∗(t)|β +∣∣G∗v∗(t)∣∣

β. (5.73)

In the first term we have used the contraction property of the heat semigroup. Here to shorten

the notation we write Ψ∗(t) for Ψθ(u∗)K2

(tK1), v∗(t) for vΨθ(u∗)K2 (tK1)(t), as well as G∗ for G

Ψθ(u∗)K2

.

Recall that tK1 = t ∧ τXK1. In particular, the construction of the process u∗ still includes all the

cutoffs, and therefore the right hand side of (5.73) depends on the Ki, although this is suppressed inthe notation.

For the second term we get due to Proposition 4.3 and Remark 4.4

E[

sup0≤t≤T

∣∣Ψ∗(t)∣∣pβ

]≤ C|θ|p0 (5.74)

for p big enough, so that this quantity can not blow up. Note that the constant in (5.74) does notdepend on the Ki.

To deal with v∗ we need to perform a calculation which is very similar to the proof of Proposition5.8 and Proposition 5.10. The difference is that we need to use the full formulas (5.9) and (5.18)instead of (5.10) and (5.20). Using the definition of the fixed point v and recalling the definitions ofthe operators G(1)

Ψ∗ and G(2)Ψ∗ from (5.41) we get

|v∗(t)|C1 ≤ |G(1)Ψ∗v

∗(t)|C1 + |G(2)Ψ∗v

∗(t)|C1 . (5.75)

In the same way as in in (5.42) we get for the term involving G(1)Ψ∗ :

|G(1)Ψ∗v

∗(t)|C1 ≤ C|g|0∫ tK1

0(tK1 − s)−

12

(|v∗(s)|C1 + s

β−12 |u0|β

)ds. (5.76)


To treat G(2)Ψ∗ write very similarly to (5.43) but using the more precise bounds (5.9) and (5.18)∣∣G(2)

Ψ∗v∗(t)∣∣C1 ≤ C|g|C2

∫ tK1

0(tK1 − s)

α2−1[

|Ψ∗(s)|α(

1 + |v∗(s)|C1 + sβ−2α

2 |u0|β + |Ψ∗(s)|2α +∣∣RΨ∗(s)

∣∣2α

)+ |X(s)|α|θ|0|RΨ∗(s)|2α + |X(s)|2α|θ|2C1

(|v∗(s)|C1 + |Ψ∗(s)|α + |u0|α

)+ |X(s)|2α|Ψ∗(s)|0 |θ|0

(|Ψ∗(s)|α + |v(s)|C1 + |u0|α

)+ |X(s)|2α|θ|0|Ψ∗(s)|α

]ds. (5.77)

We observe that all the terms on the right hand side of (5.77) except for |v∗(s)|C1 are bounded bypowers of the constants Ki and |v∗(s)|C1 only appears linearly. Thus the Gronwall Lemma 5.7gives sup0≤t≤T |v∗(t)|C1 ≤ C for a finite deterministic constant C that only depends on the Ki and|u0|β . In particular for all K1,K2,K3 > 0 we can construct u∗ up to time T . Furthermore, up toσ3K1,K2,K3

the process u∗ solves (3.3).We want to show that by choosing the constants Ki large enough it is possible to guarantee that

σ3K1,K2,K3

≥ T . Let us start by showing that limK3→∞ σ3K1,K2,K3

≥ τ|Ψ|CαK2

∧ τXK1= σ2

K1,K2i.e.

that |R|2α cannot explode for bounded Ψ and X .Now by Proposition 4.8 we know that for p satisfying (4.58)

E[∥∥Rθ(u∗)∥∥p

C([0,t∧τXK1];ΩC2α)

]≤ C(1 +Kp

1 )|θ|C1 |||u∗‖pp,α,t, (5.78)

where

|||u∗|||p,α,t = E[

supx1 6=x2,s1 6=s2≤t

|u∗(s2, x2)− u∗(s1, x1)|p(|s1 − s2|α/2 + |x1 − x2|α

)p]1/p

, (5.79)

i.e. the ||| · |||p,α-norm evaluated up to time t.Let us derive a bound on |||u|||α,p. First of all using (5.73) – (5.77) and noting that on the right

hand side of (5.77) all the terms except for those involving v∗ or RΨ∗ are bounded by powers of theconstants K1,K2 and that no term involving any product of norms of v∗ or RΨ∗ appears one canwrite for any p ≥ 1

E[

sup0≤s≤t∧σ2

K1,K2

|v∗(s)|pC1

]≤ C E

[(sup

0≤s≤t∧σ2K1,K2

∫ s

0(s− τ)

α2−1(1 + τ

β−2α2 + |v∗(s)|+

∣∣Rθ(u∗)(s)∣∣)dτ)p] (5.80)

≤ C∫ t

0(t− τ)

α2−1(1 + s

β−2α2 + E

[|v∗(s ∧ σ2

K1,K2)|pC1

]+ E

[∣∣Rθ(u∗)(s ∧ σ2K1,K2

)∣∣p2α

])dτ.

Here the constant C depends on |g|C3 , |θ|C1 , p, |u0|β,K1, and K2 but not on K3. Repeating thesame calculation as in the second half of the proof of Proposition 5.10 which is essentially anexploitation of the bound |S(t)− 1|C1→C0 ≤ Ct1/2 one gets the bound

E[‖v∗‖p

C1/2([0,t∧σ2K1,K2

],C)

](5.81)

≤ C∫ t

0(t− s)

α2−1(1 + s

β−2α2 + E

[|v∗(s ∧ σ2

K1,K2)|pC1

]+ E

[∣∣Rθ(u∗)(s ∧ σ2K1,K2

)∣∣p2α

])ds.

ROUGH BURGERS 39

Noting that according to (5.74) we know that

|||u∗|||p,α,t ≤ C(

1 + |θ|p0 + E[

sup0≤s≤t∧σ2

K1,K2

|v∗(s)|pC1 + ‖v∗‖pC1/2([0,t∧σ2

K1,K2],C)

]), (5.82)

we get using (5.78), that for p large enough to satisfy (4.58)

|||v|||pp,1,1/2,t ≤ C∫ t

0(t− s)

α2−1(1 + s

β−2α2 + |||v|||pp,1,1/2,s

)ds, (5.83)

where|||v|||pp,1,1/2,t = E

[sup

0≤s≤t∧σ2K1,K2

|v(s)|pC1 + |v|C1/2([0,t∧σ2K1,K2

],C)

]. (5.84)

So the Gronwall lemma 5.7 gives a uniform bound on this quantity for all t ≤ T . Thus using(5.78) as well as (5.82) one more time, we can deduce that

supK3

E[∥∥Rθ(u∗)∥∥p

C([0,T∧σ2K1,K2

];ΩC2α)

]≤ C(K1,K2) <∞. (5.85)

This implies by Markov inequality that

P[τ|R|2αK3

≤ T]

= P[∥∥Rθ(u)

∥∥C[0,t];ΩC2α)

≥ K3

]≤ C

Kp3

. (5.86)

This gives the desired result concerning non-explosion of RΨ before σ2K1,K2

. But then as theexpectation of the p-th moment of the α-Holder norm of Ψθ(u) is controlled by |θ|p0 and as we knowa priori that the stochastic convolution cannot blow up we see that we can remove σ2

K1,K2as well.

This finishes the proof of global existence.Uniqueness up to the stopping time σ3

K1,K2,K3follows from the construction as a fixed point in

Proposition 5.10. Then by the same argument as above we can remove the stopping times and obtainuniqueness. This finishes the proof.

Proof. (of Theorem 3.6) For ε > 0 the approximation uε is the unique solution to the fixed pointproblem

uε(t) = S(t)u0 +

∫ t

0S(t− s) g

(u(s)

)∂xu(s) ds+

∫ t

0S(t− s) θ

(uε(s)

)d(W ∗ ηε)(s)

= U(t) +Guε(t) + Ψθ(uε)ε (t). (5.87)

Of course, as uε is smooth in the space variable x there is no need for rough path theory to define theintegral involving g. Nonetheless we can view this integral as a rough integral. To be more preciseas a reference rough path one chooses

Xε(t, x, y) =

∫ y

x

(Xε(t, z)−Xε(t, x)

)dzXε(t, z). (5.88)

As Xε is a smooth function of the space variable x this integral can be defined as a usual Lebesgueintegral. Note that by Proposition 4.10 Ψ

θ(u)εε is indeed an Xε controlled rough path with decompos-

tion given asδΨθ(uε(t, x, y) = θ(uε(t, x)) δXε(t, x, y) +Rθ(uε)ε (t, x, y). (5.89)

As Xε is a smooth function this decomposition is by no means unique but it has the advantage thatProposition 4.10 provides moment estimates on the remainder, that are uniform in ε.

Introduce the stopping time

τK1,K2,K3,K4,ε = σ4,εK1,K2,K3,K4

∧ σ4,εK1,K2,K3,K4

, (5.90)


where σ4,εK1,K2,K3,K4

is defined as σ3K1,K2,K3

stopped when ‖u‖Cα/2,α exceeds K4 and is σ3,εK1,K2,K3

is defined analogously with respect to the norms derived from uε. Sometimes, we will use theshorthand τε.

We will show first, that before τε the process uε coincides with the solution to the cut-off fixedpoint problem

uε(t) = S(t)u0 + vχK5(Ψε) +

∫ t

0S(t− s) θ

(uε(s)

)d(W ∗ ηε)(s). (5.91)

Here the fixed point vχK5(Ψε) is defined as in Proposition 5.8. (In particular, we use the cutoff at

K5 = CK4 +K2 +K3 to define the cutoff of Ψε).As Ψθ(uε) is also a smooth function of the space variable x there is no need for rough path theory

in order to define the integral that determines the fixed point map in Proposition 5.8. But actually,the two integrals coincide, because in the approximation∑

i

pt−s(y − xi)g(ui)δui,i+1 + pt−s(y − xi)g′(ui)θ(ui)Xε(xi, xi+1)θ(ui)T (5.92)

the first terms converge to the usual integral∫ 1

0 pt−s(y−x)g(u(x)

)∂xu(x) dx and the sum involving

the iterated integrals converges to zero due to Xε ∈ ΩC2. In particular, all the bounds derived forthe fixed point apply in the present context. So due to uniqueness of the smooth evolution (5.91)holds.

In Section 4 we have already derived uniform bounds for the quantities appearing in this decom-position. First of all, for the derivative processes we have

|||θ(u)− θ(uε)|||p,α ≤ CK4|θ|C2 |||u− uε|||p,α. (5.93)

From (4.25) and (4.37) we have

E[∥∥Ψθ(u) −Ψθ(uε)

ε

∥∥pC([0,T ];Cα)

]≤ Cεγ |||θ(u)|||pp,α + CT κ|||θ(u)− θ(uε)|||pp,0≤ Cεγ + CT κ|||u− uε|||pp,0, (5.94)

where κ is as defined in Proposition 4.3 and γ is as in (4.36). In the Gaussian case θ = 1 this boundsimplifies to

E[∥∥X −Xε

∥∥pC([0,T ];Cα)

]≤ Cεγ . (5.95)

Using (4.26) and (4.38) we obtain the same bounds for the C([0, T ];Cα) norm replaced by theCα/2([0, T ];C) norm.

From (4.61) and (4.77) we get

E[∥∥Rθ(u)−Rθ(uε)ε

∥∥pC([0,τK1,K2,K3,ε

];ΩC2α)

]≤ C

(1 +Kp

1

)‖θ(uε)‖pp,αεγ + CT κ

(1 +Kp

1

)‖θ(u)− θ(uε)‖pp,α. (5.96)

Here we assume that κ is as defined in (4.59) and γ as in (4.76). Note that these assumptions on κand γ imply that the conditions needed above are automatically satisfied.

For the terms in the integral involving g we get using Proposition 5.10 that for T small enough

E[‖v − vε‖p

C([0,T∧τK1,K2,K3,K4,ε],C1)

]≤ C E

[∆

Ψθ(u),Ψθ(uε)ε

], (5.97)

and

E[‖v − vε‖p

C1/2([0,T ],C0)

]≤ C E

[∆

Ψθ(u),Ψθ(uε)ε

], (5.98)

ROUGH BURGERS 41

where

∆Ψθ(u),Ψ

θ(uε)ε

= ‖X −Xε‖CαT + ‖X−Xε‖ΩC2αT

+ ‖Ψθ(u) −Ψθ(uε)ε ‖CαT

+ ‖θ(u)− θ(uε)‖CαT + ‖Rθ(u) −Rθ(uε)ε ‖ΩC2αT∧τK1,K2,K3,eps

+ |u0 − uε|β.(5.99)

Thus summarising for p big enough we get the bound

E[|u− uε‖p

Cα/2,αT∧τε

]≤ Cεγ,p + CT κE

[|u− uε‖p

Cα/2,αT∧τε

]+ E

[|u− uε|pβ

], (5.100)

where we use the notation ‖u−uε‖Cα/2,αT∧τε= ‖u−uε‖Cα/2,α[0,T∧τK1,K2,K3,ε

]×[0,1]. Here the constant

depends on the Ki but not on ε. We have included the dependence on different initial data in thebound, although we start the two evolutions at the same initial data, because we will use this boundin an iteration over different time intervals.

For T small enough the prefactor of the term involving u− uε on the right hand side of (5.100) isless than 1

2 and this term can be absorbed into the term on the left hand side. We then have for Tsmall enough

E[|u− uε‖p

Cα/2,αT∧τε

]≤ Cεγp + CE

[|u0 − u0,ε|pβ

]. (5.101)

Since we start off with the same initial condition, we get convergence of the cutoff approximationsbefore T . To get the same result for arbitrary times we iterate this procedure using u(T ) and uε(T )as new initial data. To derive bounds on their difference in Cβ (not Cα!) we use the following trick.The solution u(T ) is a rough path controlled by X(T ) and we have

δu(T, x, y) = θ(u(t, x))δX(t, x, y) +Rθ(u)(t, x, y), (5.102)

and similarly for uε. Thus we have

|u(T )−uε(T )|β ≤ |θ||X(T )−Xε(T )|β+|θ|C1 |u(T )−uε(T )|0K+∣∣Rθ(u)−Rθ(uε)ε

∣∣ΩCβ

. (5.103)

We have already bounded all the quantities on the right hand side above so that we can iterate theargument. Note here that the bound on the Gaussian rough path holds for every exponent less than 1

2and thus in particular for β. The iteration gives us

E[‖u− uε‖p

Cα/2,αT∧τε

]→ 0 (5.104)

for arbitrary T and for any choice of the Ki (but not uniformly in the Ki). Now to conclude weonly need to remove the stopping times. To this end note that for δ < 1 due to the definition of thestopping time τK1,K2,K3,K4,ε

P[‖u− uε‖Cα/2,α,T ≥ δ] ≤P

[‖u− uε‖Cα/2,α,T∧τK1,K2,K3,K4,ε

≥ δ]

+ P[σ3,εK1,K2,K3,K4

≤ T]

+ P[σ3K1,K2,K3,K4

≤ T]

(5.105)

Now as in the proof of global existence by choosing the Ki sufficiently large the second probabilitiesfor the stopping times to be less than T can be made arbitrarily small . Then by choosing ε smallenough the first probability can be made arbitrarily small as well. This finishes the argument.


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MARTIN HAIRER, MATHEMATICS DEPARTMENT, UNIVERSITY OF WARWICK

HENDRIK WEBER, MATHEMATICS DEPARTMENT, UNIVERSITY OF WARWICK

E-mail address: [email protected], [email protected]

rough burgers-like equations with multiplicative …space of stochastic processes to deal with the...

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